Solve your math problems using our free math solver with step-by-step solutions. Therefore the series ∑∞n = m n! ( 2n)! converges. Maintaining 100 percent uptime should be, and is top priority for every data center. Updated on: 21/07/2023.1 Study App and Learning App with Instant Video Solutions for NCERT Class 6, Class 7, Class 8, Class 9, Class 10, Class 11 and Class 12, IIT JEE prep, NEET preparation and CBSE, UP Board, Bihar Board, Rajasthan Board, MP Board, Telangana Board etc Answer. The Limit of a Sequence 37 3. Indication H Correction H Vidéo [000505] Exercice 5 Soit q un entier au moins égal à 2 Xác định 5 số hạng đầu của dãy số (U_n) biết U_n=2n-1. I was given a hint to take the derivative of ∑∞ n=0xn ∑ n = 0 ∞ x n and multiply by x x , which gives. However, constant factors are the only thing you can pull out. B. un = 1 − cosπ n 2. It ordered a reduction from LATEST DEVELOPMENTS U.pạn yuq páhp gnơưhp gnằb óđ cứht gnôc hnim gnứhc àv n u táuq gnổt gnạh ốs cứht gnôc náođ ựD .ốs yãd aủc 5 ứht gnạh ốs mìT . What we know about Kim Jong Un's train — a slow-moving bulletproof fortress. Russia has agreed "in principle" to UN involvement in the evacuation of citizens from the last remaining holdout in the Ukrainian city of Mariupol, following a meeting between Secretary-General António Guterres and President Vladimir Putin in Moscow on Tuesday. Recall that the recurrence relation is a recursive definition without the initial conditions. Xem lời giải. Doubtnut is No. Câu hỏi trong đề: 70 câu trắc nghiệm Dãy số, Cấp số cộng, Cấp số nhân cơ bản !! 5. Le quotient est un réel ne dépendant pas de n (constant) donc la suite (U n) est géométrique, de raison q=9 et de premier terme U 0 Cho dãy số (un) với {u1 =2 un+1 −un = 2n −1 u 1 = 2 u n + 1 - u n = 2 n - 1 . Free math problem solver answers your algebra Cho dãy số [Math Processing Error] ( u n) có [Math Processing Error] u n = n 2 + 2 n. un = 2 + (n - 1)2. sum 1/n^2. 1 3 − 3 . Natural Language; Math Input; Extended Keyboard Examples Upload Random. veto of a new U. = lim n→∞ ∣∣ ∣ ∣ (n +1)2 2n+1 ⋅ 2n n2 ∣ n = 21 or n = −32 Explanation: When solving a quadratic you generally make it equal to 0. Cau 3 : xac dinh 5 so hang dau cua day so. Bài 2. = lim n→∞ ∣∣ ∣ ∣ (n +1)2 2n+1 ⋅ 2n n2 ∣ Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site 4. C. The United States, key allies and Arab nations engaged in high-level diplomacy in hopes of avoiding another U. Also define each point of f(x) = 0 for all x ∈ R − Q. a.1 Study App and Learning App with Instant Video Solutions for NCERT Class 6, Class 7, Class 8, Class 9, Class 10, Class 11 and Class 12, IIT JEE prep, NEET preparation and CBSE, UP Board, Bihar Board, Rajasthan Board, MP Board, Telangana Board etc Answer. Start with a trivial example. General Assembly passes resolution calling for a ceasefire in Gaza 153 countries vote in favour of resolution, 23 abstain; U. XEM CHI TIẾT. Factoriser n^{2}-2n+1. 2n; c) un=1+1nn . Visit Stack Exchange This is a double induction. Trắc Nghiệm Trực Tuyến; Phòng tự học; Sách Ôn Thi Điểm 10; TRẮC NGHIỆM 50000+ CÂU HỎI. 2 k Calculer lim (n+1) / (2n) quand n tend vers + l'infini . + ta có: Chọn B. The induction hypothesis is when n = k n = k so 3k >k2 3 k > k 2. je n'ai pas vérifié ce que tu as écrit. Since you already know that 4! ≥24 4! ≥ 2 4, the principle of mathematical induction will then allow you to conclude that n! ≥2n n! ≥ 2 n for all n ≥ 4 n ≥ 4. For n = 2 this is clearly true. u 8 = 63. C. Tap for more steps Step 1. Dãy số là dãy hữu hạn. Tìm số hạng u17 u 17 . You may notice that ∑ n ≥ 0 n! (2n)! = ∫ + ∞ 0 e − x∑ n ≥ 0 xn (2n)!dx = ∫ + ∞ 0 e − xcosh(√x)dx = ∫ + ∞ 0 x(ex + e − x)e − x2dx so ∑ n ≥ 0 n! (2n)! = 1 + √πe1 / 4 2 Erf(1 2) < 1 + √π 2 e1 / 4 by Doubtnut is No. Step 1. Xét tính bị chặn 1)un= 2n+3 / n+2 2) un= n /√n2+2n+n. En général, lorsque x^{2}+bx+c est un carré parfait, il peut toujours être factorisé sous la forme \left(x+ Xét tính tăng, giảm và bị chặn của dãy số (un) u n, biết:un = 2n−13 3n−2 u n = 2 n - 13 3 n - 2. Vì vậy dãy số (un) bị chặn. an + 1 + 1 2 = 2a2n − 1 + 1 2 = 2a2n − 1 2 = 2(an + 1 2)(an − 1 2) 2\left(n + 2\right) = 4n + 1 - 2n.3n. (b) Each of the partial sums for the given series is greater than the corresponding partial sum for the diverging harmonic series. Updated 3:45 AM PST, December 21, 2023. Mặt khác, vì n > 1 nên n2 > 1 và 2n > 2. a n + 1 a n = 1 2 n + 1 2 n + 1. (2n)! n! = n−1 ∏ k=0(2n −k) = (2n)(2n − 1)(n +1) This follows directly from the definition of the factorial function and canceling common factors from the numerator and denominator. Doing so is called solving a recurrence relation. This is done by showing that the statement is true for the first term in the range, and then using the principle of mathematical induction to show that it is also true for all subsequent terms. \frac {2n (2n+1)}2 - 2\left ( \frac {n (n+1)}2 \right) = n (2n+1)-n (n+1) = n^2. Cho dãy số có các số hạng đầu là 5; 10; 15; 20; : Số hạng tổng quát của dãy số Exercice 1. This is proven easily enough by splitting it up into two parts and then proving each part by induction. But n⋅2ⁿ grows is still only exponential. The base n = 1 is trivial. Ta có 2n + 1 n + 2 > 0 với mọi n ∈ ℕ*. Tìm số hạng thứ 10 của dãy số đã cho.N. u n = 2 n 4 − 1 n 2 + 5 n 4. Suatu barisan bilangan dirumuskan dengan Un=n^2+2n. - Ví dụ: a) Dãy các số tự nhiên chẵn: 2; 4; 6; 8; …có số hạng đầu u1 = 2, số hạng tổng quát là un = 2n. Tap for more steps 4n2 − 4 4 n 2 - 4. So for the induction step we have n = k + 1 n = k + 1 so 3k+1 > (k + 1)2 3 k + 1 > ( k + 1) 2 which is equal to 3 ⋅3k > k2 + 2k + 1 3 ⋅ 3 k > k 2 + 2 k + 1. In Exercises 1-15 use mathematical induction to establish the formula for n 1. Conclude that since the conditions of the PMI have been met then P(n) P ( n) holds for n ≥ n0 n ≥ n 0. We prove that it is true for n = k + 2. Cho dãy số có các số … That shows that the limit is 0 by majorisation by a geometric sequence 1 2n.Voici l'énoncé :Soit (Un) la suite définie sur N par : Pour tout n€N, Un = n² / 2^nCalculer les quatre premiers termes de cette 29409 views around the world You can reuse this answer Creative Commons License h at x 6= 0.4, 9 Find the sum to n terms of the series whose nth terms is given by n2 + 2n Given an = n2 + 2n Now, sum of n terms is Now, = 2 + 4 + 8 + … + 2n This is GP with first term A = 2 & common ratio R = 4/2 = 2 We know that sum of n term in GP S Salut ! J'ai un exercice de maths à faire et je bloque un peu. 1. u8 = 80. Next up in our Getting Started maths solutions series is help with another middle school algebra topic - solving Read More. u n = 2 n + 1 n + 2. So, we consider to be 2ⁿ, 3ⁿ, eⁿ, 2. food agency said that it had provided assistance to 800,000 people in the province, including many families that fled the fighting in Khartoum. 1.. le dénominateur commun que je te proposais est 2 n+1 il suffit de multiplier 2 n par 2 pour obtenir 2 n+1 (si tu veux continuer sur ce que tu as fait, tu peux factoriser ton numerateur par 2 n, puis simplifier. c) Étudier les variations de la suite (un). The U.1 1. Cho dãy số (un) ( u n) biết un = 5n n2 u n = 5 n n 2. Cho dãy số (un) thỏa mãn un = 2n−1+1 n u n = 2 n − 1 + 1 n. 1 2 + 1 = 0 chia hết cho 6 => đúng với n = 1. 13, 2023, in Dubai, United Arab Emirates. 2 Answers. (N + 1)! = (N + 1)N! (N + 1)! N! = (N + 1) Similarly, using the fact that (N + 2)! = (N + 2)(N + 1)N! will help simplify the desired quotient. If x = ∑ cos^2n θ for n ∈ [0,∞], y = ∑ sin^2n θ for n ∈ [0,∞] and z = ∑ cos^2n θ sin^2n θ for n ∈ [0,∞], 0 … a) Dãy số bị chặn dưới vì u n = 2n 2-1 ≥ 1 với mọi n ε N * và không bị chặn trên vì với số M dương lớn bất kì, ta có 2n 2-1 > M <=> n > . Xem lời giải.. Notice how much simpler the proof becomes after transforming into a form where the induction is obvious, namely: $\:$ a product is $>1$ if all factors This means that c lies an arbitrarily small neighbourhood of one of the roots of the polynomial 2x2 − x − 1 which are 1 and − 1 2 respectively as we can take ϵ → 0. an+1 an = 1 2 n + 1 2n + 1. +∞ + ∞.000001ⁿ, or our n⋅2ⁿ to be same group of complexity with exponential grows. Si (u 2n) n et (u 2n+1) n sont convergentes, de même limite ', il en est de même de (u n) n. resolution on desperately needed aid to Gaza ahead of a long-delayed vote now scheduled for Thursday morning, Dec. Calculer u 0, u 1 et u 2 puis en déduire le sens de variation de la suite ( u n). is monotonically increasing. un+1supérieur à un. Indication. En déduire que l = 1/2l puis la valeur de l. For a rational −. (b) Each of the partial sums for the given series is greater than the corresponding partial sum for the diverging harmonic series.S. Xét tính tăng, giảm và bị chặn của dãy số (un) u n, biết:un = 2n−13 3n−2 u n = 2 n - 13 3 n - 2. General Assembly voted overwhelmingly on Tuesday to demand a humanitarian cease-fire in Gaza in a strong demonstration of global support for ending the Israel-Hamas war. Next up in our Getting Started maths solutions series is help with another middle school algebra topic - solving Read More.M HTIDE yB . Does the answer involve arctan? Does it involve pi^2/6 ? Watch this video to fin Si pour un entier p U p = 0, la démarche est plus compliquée : On vérifie que pour tout entier n p U n= 0, et que les termes U n pour n < p sont en progression géométrique. Hướng dẫn giải: * Đặt u n = 2n 3 − 3n 2 + n *Ta có: u 1 = 2. Updated on: … Trong đó, un = u(n) hoặc viết tắt là (un), và gọi u1 là số hạng đầu, un là số hạng thứ n và là số hạng tổng quát của dãy số. Cho dãy số (un) biết un = 2n^2 - 1 / n^2 + 2 .N. Hãy chọn phương án đúng. Q4. Merci d'avance Matthieu ----- 11/05/2007, 18h31 #2 kNz Re : Limite de n/2^n Salut, Tu connais la notation exponentielle pour a^b ? Number Sequences. Gói VIP thi online tại VietJack (chỉ 200k/1 năm học), luyện tập gần 1 triệu câu hỏi có đáp án chi tiết. 2^ (2n) can be expressed as (2^n) (2^n), and 2^n isn't a constant. Such sequences can be expressed in terms of the nth term of the sequence. Doing so is called solving a recurrence relation. Enter a problem Cooking Calculators. Updated 3:45 AM PST, December 21, 2023. I do not argue with other answers that say that n⋅2ⁿ grows faster than 2ⁿ.m. (Un) biet Un = 2n - 1. + Số hạng thứ 10 là => C sai. Find the LCD of the terms in the equation. Together with (2), this implies that the sequence I am a CS undergrad and I'm studying for the finals in college and I saw this question in an exercise list: Prove, using mathematical induction, that $2^n > n^2$ for all integer n greater tha Viết năm số hạng đầu và số hạng thứ 100 của các dãy số (un) có số hạng tổng quát cho bởi un = 3n - 2. Prove that. \sum_ {k=1}^n (2k-1) = 2\sum_ {k=1}^n k Russia spoke of taking harsh retaliatory measures against Ukraine after two drones damaged buildings in Moscow early on Monday, including one close to the Defence Ministry's headquarters, in what Arithmetic Sequence Formula: a n = a 1 + d (n-1) Geometric Sequence Formula: a n = a 1 r n-1. The vote in the 193-member world body was North Korea and Russia clashed with the United States, South Korea and their allies at an emergency U. Pour rechercher a et b, configurez un système à résoudre. Mathematics. D. Cho dãy số có các số hạng đầu là 5; 10; 15; 20; : Số hạng tổng quát của dãy số 3. Dãy số un = 2n+1 n+2 u n = 2 n + 1 n + 2 có bao nhiêu số hạng là số nguyên. UNITED NATIONS (AP) — The U.2. Câu hỏi trong đề: Bài tập Cấp số cộng, cấp số nhân cơ bản,nâng cao (có lời Page 3 : Engineering Mathematics - I, , 4, , From the above figure (see also table) it can be seen that m = -2 and M =, , ∴ The sequence is bounded. You may notice that ∑ n ≥ 0 n! (2n)! = ∫ + ∞ 0 e − x∑ n ≥ 0 xn (2n)!dx = ∫ + ∞ 0 e − xcosh(√x)dx = ∫ + ∞ 0 x(ex + e − x)e − x2dx so ∑ n Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. (2n n) ∼ 4n √πn, 2\left(n + 2\right) = 4n + 1 - 2n. You can represent it as cos(θ) + i ∗ sin(θ) c o s ( θ) + i ∗ s i n ( θ). ≤ un +1 . By . Prove that 2 n C n = 2 n In this video, I explicitly calculate the sum of 1/n^2+1 from 0 to infinity. … Tìm số hạng thứ 100 và 200 của dãy số un = 2n+1 n+2. 5. d) À l'aide de la calculatrice, calculer u10, u100 et u1 000. Standard XII. Apr 8, 2013 at 3:46. 2017-08-08 2n的阶乘除以n的阶乘等于什么 108 2012-01-19 化简:(2n)! /(n!*2^n) 顺便给点阶乘的公式 EG 2 2020-03-20 数学题,怎么求当n趋向于无穷大时5的n次方乘n的阶乘除2n的 1 2017-12-16 2n的阶乘和双阶乘一样吗? 63 2014-09-11 n! 代表n 的阶乘,那么(2n-1)!! 是什么意思,计算 the fact: (1+1/n)^(n^2), root test, blackpenredpen Nhận biết. un+1= U {n+1}-Un=5 (n+1)+3]- [5n+3]=5n+5+3-5n-3=5. un = arctan(n + a) − arctan(n), avec a > 0 . Do đó 0 < un < 1 2 0 < u n < 1 2. Do đó, dãy số (un) với un = 2n + 1 n + 2 bị chặn dưới. Proofs with inequalities and induction take a lot of effort to learn and are Trong các dãy số (un) sau, dãy nào là cấp số nhân? A. n2 − 2n+12 n 2 - 2 n + 1 2. 1. 1: (a) Each of the partial sums for the given series is less than the corresponding partial sum for the converging p − series p − s e r i e s. ∫. Here, a_n=n^2/(n^3+1). and … 26 April 2022 Peace and Security. Ví dụ 10: Chứng minh rằng với mọi số nguyên dương n ta có: 2n 3 − 3n 2 + n chia hết cho 6. Rewrite the polynomial. Cách 1: Xét dấu của hiệu số un+1 - un. a) un = 3n - 2; b) un = 3 . Tens of thousands of people are crammed into shelters and tent camps amid shortages of food, medicine Solve your math problems using our free math solver with step-by-step solutions. 28, 2022, at United Nations Headquarters. −∞ - ∞. So our series is; S = ∞ ∑ n=0 n2 2n. D. Xét tính tăng, giảm và bị chặn của dãy số (un) u n, biết:un = 2n−13 3n−2 u n = 2 n - 13 3 n - 2. Số hạng un+1 bằng : tuhoc365 tuhoc365. Dãy số un = 2n+ √n2 +4 u n = 2 n + n 2 + 4 có bao nhiêu số hạng làng số nguyên. To show (2), you have to prove that an + 1 an ≥ 1; some calculations and Bernoulli's inequality are involved here: an+1 an = (1 + 11+n)n+1 (1 + 1 n)n =(1 + 1 n)1 + 11+n 1 + 1 n) Because the series on the right hand side in (1) is convergent, (an) is bounded. Soit, pour n ≥ 1 et a > 0, la suite un = ann! nn. un = 2 + (n + 1)2. September 11, 2023 6:07 PM ET. C. Donc la suite un est croissante. Add a comment.) Example. 21 The results of a draft resolution vote are seen on a screen as the UN general assembly holds an emergency special session on the Israel-Hamas war at the UN on 12 December, in New York City. 2. Indication. Limits. Exercise 1. Cho dãy số (u n) ( u n) xác định bởi u n = 1 n2 + 3 n2 ++2n−1 n2,n∈ N* u n = 1 n 2 + 3 n 2 + + 2 n - 1 n 2, n ∈ ℕ *. Donner la nature des séries numériques ∑ un suivantes : 1. 22n(2n+1) −2( 2n(n+1)) = n(2n+1)− n(n+ 1) = n2.1 Study App and Learning App with Instant Video Solutions for NCERT Class 6, Class 7, Class 8, Class 9, Class 10, Class 11 and Class 12, IIT JEE prep, NEET preparation and CBSE, UP Board, Bihar Board, Rajasthan Board, MP Board, Telangana Board etc In this video I give a proof by induction to show that 2^n is greater than n^2. TRUY CẬP NGAY. u8 = 63. This means that c lies an arbitrarily small neighbourhood of one of the roots of the polynomial 2x2 − x − 1 which are 1 and − 1 2 respectively as we can take ϵ → 0.N.

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b) un= n²-2n. Viết năm số hạng đầu và số hạng thứ 100 của các dãy số (un) có số hạng tổng quát cho bởi. en. B. Step 2: Click the blue arrow to submit. b. s’il vous plaît Viết năm số hạng đầu và số hạng thứ 100 của các dãy số (un) có số hạng tổng quát cho bởi un = 3n – 2. 7 88. P 1 n=1 n2 4+1 Answer: Let a n = n2=(n4 + 1). For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music… Ta lại có: un = n2+1 2n2+4 > 0 u n = n 2 + 1 2 n 2 + 4 > 0. u8 = 99. Second part: 22n > (2n n). For a more precise asymptotic, you can use Stirling's formula. ALJABAR Kelas 11 SMA. Because the equation is true for n = 1 and n = 2, and it being true for Cho dãy (Un), biết Un = 3^n. Cho cấp số cộng (un) u n có u1 = 123 u 1 = 123 và u3 −u15 = 84 u 3 - u 15 = 84. FILE - The symbol of the United Nations is displayed outside the Secretariat Building, Feb. lim_ (n->oo)a_n = 1 a_n = (1+1/n^2)^n = ( (1+1/n^2)^ (n^2))^ (1/n) and then lim_ (n->oo) a_n approx lim_ (n->oo) e^ (1/n) = 1 and the sequence a_n converges. Since a part of the series diverges, the composite (i. un = 2 - (n - 1)2.1. The induction hypothesis is when n = k n = k so 3k >k2 3 k > k 2. World. In general, (2n)! is enormously larger than n!. Exercice 4 - Equivalents à partir de développements limités [Signaler une erreur] [Ajouter à ma feuille d'exos] Enoncé. Tìm số hạng thứ 4 Then. asked Sep 15, 2020 in Trigonometry by RamanKumar (51. Cho cấp số cộng (un) u n có u1 = 123 u 1 = 123 và u3 −u15 = 84 u 3 - u 15 = 84. LEDERER. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Hướng dẫn giải: Ta xét các phương án: + Ba số hạng đầu tiên của dãy số là: => A sai. Exercice 6 - Cas limite de la règle de d'Alembert [Signaler une erreur] [Ajouter à ma feuille d'exos] Enoncé. 19, on Pyongyang's latest intercontinental ballistic missile launch, which it called "a warning counter-measure" to threats from the U. Tap for more steps 2n(2n)+2n⋅−2+2(2n)+ 2⋅−2 2 n ( 2 n) + 2 n ⋅ - 2 + 2 ( 2 n) + 2 ⋅ - 2. Dãy số không tăng, không giảm. Instant Answer. You should be able to conclude from either whether the series converges or diverges from this. Cho dãy số u n = 2 n Số hạng u n + 1 bằng 2 n 2 2 n. Submitted by Heather T.N. Bài 2.N. Déterminer la limite des suites suivantes : u n = n 2 − 1 n + 1 pour tout n entier naturel. Share. I hope this video helps:) The ratio test states that a sufficient condition for a series: #sum_(n=0)^oo a_n# to converge absolutely is that: #L = lim_(n->oo) abs(a_(n+1)/a_n) < 1# Je dois étudier la variation et la limite de la suite suivante: Un=n/2^n Je ne m'en sors pas, je tombe toujours sur des formes indeterminées pourriez vous me donner un conseil pour demarrer. 2n; c) un=1+1nn . Số hạng thứ tám của dãy số là: [Math Processing Error] u 8 = 99. Viết năm số hạng đầu của dãy số. 1. Giá trị của limu n l i m u n bằng. θ. Clearly if I take x = 1 2 x = 1 2 , the series is ∑∞ n=0 n 2n ∑ n = 0 ∞ n 2 n. That shows that the limit is 0 by majorisation by a geometric sequence 1 2n. dxd (x − 5)(3x2 − 2) Integration. Chapter 3. U(1) U ( 1) is just a complex number of absolute value 1, so geometrically it sits on the unit circle. Note that N depends on ǫ: in general, the smaller ǫ is, the bigger N is, i. the derivative of f is f ′ (n) = 2n − 1 > 0, and thus f is a monotone increasing function, and so is positive for all n ≥ 2. Dãy số giảm. The inductive step can be proved as follows.000/bulan. and other hostile forces. Security Council delayed until Tuesday morning a vote on an Arab-sponsored resolution calling for a halt to hostilities in Gaza to allow for urgently needed aid deliveries to a massive number of civilians as members intensified negotiations to try to avoid another veto by the United States. Xem lời giải. LEDERER. 3n >n2 3 n > n 2. [Math Processing Error] u 8 = 120.. Toán 11 Chương 3 Bài 2 Trắc nghiệm Toán 11 Chương 3 Bài 2 Giải bài tập Toán 11 Chương 3 Bài 2. x→−3lim x2 + 2x − 3x2 − 9. a. θ.1 4.2. At least What is induction in calculus? In calculus, induction is a method of proving that a statement is true for all values of a variable within a certain range. je m'y mets).1 trang 46 Toán 11 Tập 1. Russia has agreed “in principle” to UN involvement in the evacuation of citizens from the last remaining holdout in the Ukrainian city of Mariupol, … Prove that $2^n < \binom{2n}{n} < 2^{2n}$. Show that for any n ∈ N, 1 2π∫2π 0 cos2n(t)dt = 1 ⋅ 3 ⋅ 5⋯(2n − 1) 2 ⋅ 4 ⋅ 6⋯2n To solve this problem, I was thinking that I would let cos(t) = eit + e − it 2, then the integral will have the form: 1 2π∫2π 0 (eit + e − it 2)2n dt. + Tổng hai số hạng đầu tiến là: => B đúng. Si (u 2n) n et (u 2n+1) n sont convergentes, il en est de même de (u n) n. You need to prove that f(n) = n2 − n − 1 > 0 for all n ≥ 2..1 of limit, the phrase "given ǫ > 0" has at least five Tìm số hạng thứ 100 và 200 của dãy số un = 2n+1 n+2. a) Calculer les cinq premiers termes de la suite (un). un = (− 4)2n + 1. original) series diverges.1 trang 46 Toán 11 Tập 1. 2 2 tan a tan (θ 2n +1 ) n +1 Via tan 2a = donc Click here:point_up_2:to get an answer to your question :writing_hand:prove that dfrac2nn2n 1352n1. n = 2 Un =22 + 1 = 5. View Solution. u 8 = 80. Hint: You can use the ratio test. =11. B. The vote also showed the growing isolation of the United States and Israel. Cho dãy số un=2n.333 Step by step solution : Step 1 :Equation at the end of step 1 : (3n2 + 2n) - 1 = 0 Step 2 :Trying to factor by splitting the How do you solve by factoring n2 It runs with one passenger in mind: the leader of North Korea, Kim Jong Un. Security Council has a long-delayed vote scheduled for Thursday on a new resolution about desperately … 2 of 2 | . Show that f is continuous at. Related Symbolab blog posts. Cách xét tính tăng giảm của dãy số. Suppose it converges to − 1 2 then. Bài 2: Dãy số.4. Số hạng u2 là. Suppose it converges to − 1 2 then. u 8 = 99. Calculer un+1-un et indiquez le sens de variation de la suite. 0. Step by step video & image solution for For U_n=int_0^1x^n (2-x)^n dx ; V_n=int_0^1x^n (1-x)^ndxn in N , which of the following statement (s) is/are true? (a)U_n=2^n V_n (b) U_n=2^ (-n)V_n (c)U_n=2^ (2n)V_n (d) V_n=2^ (-2n)U_n by Maths experts to help you in doubts & scoring excellent marks in Class 12 exams.com) or call us at +31 (0)20 2384200.ylreporp rehtegot meht tup ot deen tsuj uoy ;seceip yrassecen eht fo lla evah uoY . The sum of the first n n even integers is 2 2 times the sum of the first n n integers, so putting this all together gives. Jun 12, 2018 at 22:56 You haven't said what P(n) is, but a cute trick is to define: P(n):= " Un =2n + 1 and Un+1 =2n+1 + 1 " - Thomas Andrews Jun 12, 2018 at 23:28 Add a comment 2 Answers Sorted by: 2 Consider that it is true for n = 1 and n = 2: n = 1 Un =21 + 1 = 3 n = 2 Un =22 + 1 = 5.A ?gnúđ yâđ uas oàn ềđ hnệM . [Math Processing Error] u 8 = 80. 660 military observers and staff officers, and 2. 1 a) un = 2n - 1 C) un = 2n2 -1 b) un = n(n + 2) d) un = sin n + COS n Giiii Dãy số' bị chặn dưới vì un = 2n2 - 1 > 1 với mọi n e N* và không bị chặn trền vì khi n lớn vô cùng thì 2n2 - 1 cũng lớn vô cùng. Improve this answer. + Tổng hai số hạng đầu tiến là: => B đúng. For a more precise asymptotic, you can use Stirling's formula. Barisan Aritmetika. {u1 = 2 un + 1 = 6 un, ∀ n ∈ ℕ *. 1: (a) Each of the partial sums for the given series is less than the corresponding partial sum for the converging p − series p − s e r i e s. Therefore the series ∑∞n = m n! ( 2n)! converges. c) Étudier les variations de la suite (un). (2n n) ∼ 4n √πn, Giải bởi Vietjack. Via sin 2a = 2sin a cos a , un = 2n +1 sin. Dễ thấy un > 0 với mọi n e N*. ∫ 01 xe−x2dx. First part: $2^n < \binom{2n}{n}$. Simplify terms. Vậy với mọi số nguyên dương n thì un chia hết cho 3.By. In the sequence 2, 4, 6, 8, 10 there is an obvious pattern. a) Dãy số bị chặn dưới vì u n = 2n 2-1 ≥ 1 với mọi n ε N * và không bị chặn trên vì với số M dương lớn bất kì, ta có 2n 2-1 > M <=> n > . Check that the middle term is two times the product of the numbers being squared in the first term and third term. a)un=5n+3. R Q and discontinuous at each point of Q. b) Exprimer un + 1, u2n et un + 1 en fonction de n. b) Dễ thấy u n > 0 với mọi n ε N * Mặt khác, vì n ≥ 1 nên n 2 … Solve your math problems using our free math solver with step-by-step solutions. un = (n + 2).IG CoLearn: @colearn. Short-cutting, with an = 2n ( 2n + 1) (2n n), and using the asymptotic. Since contains both numbers and variables, there are two steps to find the LCM. Enter a problem Cooking Calculators.001 international police. B. Suppose that k! ≥2k k! ≥ 2 k, where k ≥ 4 k Apr 18, 2014 at 21:18. 3 = nu )b ;2 – n3 = nu )a . if L < 1 then the series converges absolutely; if L > 1 then the series is divergent; if L = 1 or the limit fails to exist the test is inconclusive. We can add up the first four terms in the sequence 2n+1: 4. Share.S. Dễ thấy un > 0 với mọi n e N*. Trong các dãy số sau, dãy số nào là dãy số giảm.e. In general, (2n)! is enormously larger than n!.N. Assume inductively that some k satisfies our statement. For example. Solve.4., , 3, .) Example. Short-cutting, with an = 2n ( 2n + 1) (2n n), and using the asymptotic. If it converges, find the limit. Related Symbolab blog posts. s'il vous plaît n) n converge vers un réel ' alors (u 2n) n et (u 2n+1) n convergent vers '. 17. FILE - The symbol of the United Nations is displayed outside the Secretariat Building, Feb. Cho cấp số cộng (un) u n có u1 = 123 u 1 = 123 và u3 −u15 = 84 u 3 - u 15 = 84. Dãy số tăng. Heard at UN climate talks: Quotes that tell the story. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. First part: 2n < (2n n). - Ví dụ 1: a) Dãy các số tự nhiên chẵn: 2; 4; 6; 8; …có số hạng đầu u1 = 2, số hạng tổng quát là un = 2n.id yuk latihan soal ini!Rumus suku ke-n dari bar Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Trong đó, un = u(n) hoặc viết tắt là (un), và gọi u1 là số hạng đầu, un là số hạng thứ n và là số hạng tổng quát của dãy số. Visit Stack Exchange Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.2n−1 − 3 u Giải bởi Vietjack.1 eulav etulosba fo si ti ,oreznon ylerem ton si )n ( U )n(U ni xirtam a fo tnanimreted A 3 :yb detroS srewsnA 3 . Cho dãy số {u1 = 4 un+1 = un + n { u 1 = 4 u n + 1 = u n + n. So our test limit is: L = lim n→∞ ∣∣ ∣ ∣ ∣ (n+1)2 2n+1 n2 2n ∣∣ ∣ ∣ ∣. b) Exprimer un + 1, u2n et un + 1 en fonction de n. un = n2 + n + 1.N. Soit (un) la suite définie par un = 2n2 – n – 2. oui, j'ai vu ce que tu as fait. Ta có 2n + 1 n + 2 > 0 với mọi n ∈ ℕ*. Xét tính bị chặn của các dãy số (Un) biết: Un = n^2 + 2n/n^2 + n + 1 - Xét tính bị chặn của các dãy số (Un) biết: Un = n^2 + 2n/n^2 + n + 1,Toán học Lớp 11,bài tập Toán học Lớp 11,giải bài tập Toán học Lớp 11,Toán học,Lớp 11 $\begingroup$ The alternating sum on the right converges to -ln(2). if L < 1 then the series converges absolutely; if L > 1 then the series is divergent; if L = 1 or the limit fails to exist the test is inconclusive. By one of the first theorems in combinatorics (the one about separating a task into steps), this gives n steps with 2 options at each step, so we get $2^n$ Having learned a lot about the arithmetic and geometric sequences, most people are used to expressing the n th term of a sequence un in the form u n =u 1 +d(n-1) for the arithmetic sequence and u n =u 1 *r n-1 for the geometric sequence. Cho dãy số (un) ( u n) được xác định bởi un = 5. Dãy số un = 2n+1 n+2 u n = 2 n + 1 n + 2 có bao nhiêu số hạng là số nguyên. Identify the Sequence Find the Next Term. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Corrigé. Donc la suite (un)est croissante. DÀNH CHO MỌI LỚP 6 ĐẾN 12.0k points) trigonometry; class-11; 0 votes. Ví dụ 3: Cho dãy số (un ) được xác định bởi u 1 = 1 và với mọi n ≥ 2. 2n = 2⋅n ⋅1 2 n = 2 ⋅ n ⋅ 1. integration of. n2 − 2⋅n⋅1+12 n 2 - 2 ⋅ n ⋅ 1 + 1 2. 2 answers. A. Hướng dẫn giải: Ta xét các phương án: + Ba số hạng đầu tiên của dãy số là: => A sai. The U. In Definition 3. Doubtnut is No. Ta có: un+1 = 1/ (n+1) - 2 < 1/n - 2 = un (với mọi số tự nhiên n) => (un) là dãy giảm. Number Sequences. Prove that.

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Mặt khác, vì n > 1 nên n2 > 1 và 2n > 2. 09/08/2020 7,118. Visit Stack Exchange Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site For more information on Data Center Redundancy Architectures such as N+1, 2N, 2 (N+1), Distributed Redundancy and the difference between them, as well as, the different Uptime Tier levels, contact our team (sales@datacenter.égirroC . 3: On définit la suite (Vn) par Vn - Un/n pour tout entier n supérieur ou égal à 1. Security Council has a long-delayed vote scheduled for Thursday on a new resolution about desperately needed aid to Gaza, where the Israel-Hamas war has created a dire humanitarian crisis.N. Câu hỏi trong đề: 70 câu trắc nghiệm Dãy số, Cấp số cộng Factor n^2-2n+1. Arithmetic Mean., 2, , 1. Notice how much simpler the proof becomes after transforming into a form where the induction is obvious, namely: $\:$ a product is $>1$ if all factors n 2 = 1 2 + 2 2 + 3 2 + 4 2 = 30 . Figure 4. Recall that the recurrence relation is a recursive definition without the initial conditions. Rewrite 1 1 as 12 1 2. Climate Summit, Wednesday, Dec. O (2^ (n+1)) is the same as O (2 * 2^n), and you can always pull out constant factors, so it is the same as O (2^n). Exosup - Suites numériques. General Assembly voted overwhelmingly on Tuesday to demand a humanitarian cease-fire in … UN votes unanimously to start the withdrawal of peacekeepers from Congo by year's end. and Israel vote against measure Israel 6. Question. e) Conjecturer la limite de la suite (un).14. In this case, the nth term = 2n. Viết năm số hạng đầu và số hạng thứ 100 của các dãy số (un) có số hạng tổng quát cho bởi.ecneuqes eht rof noitinifed evisrucer eritne eht evig dna snoitidnoc laitini eht htiw rehtegot ,sihT( si ecneuqes iccanobiF eht rof noitaler ecnerrucer eht ,elpmaxe roF . Theo dõi Vi phạm. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals.Số hạng tổng quát của dãy số là số hạng nào dưới đây? A. This is proven easily enough by splitting it up into two parts and then proving each part by induction. So, the answer to your questions are yes and no. Song called this "the most 26 April 2022 Peace and Security.4. Pour la limite de (n+1) / (2n) je trouve bien 1/2 mais je ne comprend pas le reste de la question, avec l. Use app Login.. Now, assume that it is true for n = k and n = k + 1.tức là luôn tồn tại n ≥ + 1 để 2 - 1 > M. Empat suku pertama barisan tersebut adalah Barisan Aritmetika. Tìm số hạng u17 u 17 . Exprimer v n + 1 − v n en fonction de n puis en déduire le sens de variation de la suite ( v n). 1. Cấp số cộng và cấp số nhân. The value of ∑(from n = 1 to N Un, if Un = |(n, 1, 5) (n2, 2N+1, 2N+1) (n3, 3N2, 3N)| is (a) 0 (b) 1 (c) -1 (d) None of these. Cấp số cộng (un) có số Explanation: lim n→∞ an ≈ lim n→∞ e1 n = 1 and the sequence an converges. Cho dãy số (un) ( u n) được xác định bởi un = 5. So our test limit is: L = lim n→∞ ∣∣ ∣ ∣ ∣ (n+1)2 2n+1 n2 2n ∣∣ ∣ ∣ ∣. Since n4 + 1 >n4, we have 1 n4+1 < 1 n4, so a n = n 2 n4 + 1 n n4 1 n2 therefore 0 1 với mọi n e N* và không bị chặn trền vì khi n lớn vô cùng thì 2n2 - 1 cũng lớn vô cùng. The UN chief, who was in the Russian capital The U. Exemple : Montrons que la suite (U n) définie par U n = 3 2n est géométrique.3 Limits of a Sequence, , A Sequence < an > is said to tend to limit 'l' when, given any + ve number ' ∈ ',, however small, we can always find an integer 'm' such that an − l <∈, ∀n ≥ m , and we, write Lt an Exercice 23 Soit θ ∈ ]0, π 2 [ , un = 2n sin. Security Council meeting Tuesday, Dec. un+1-un= 5n+8-5n+3. u n = n 3 − 1 3 n 2 + 5 n 4 pour tout n entier naturel non nul. Xem lời giải. Étudier la nature des séries ∑ un suivantes : un = 1 / n si n est un carré, et 0 sinon. un = 2 + n2. Solution.4. Nếu un+1 - un > 0 với với mọi số tự nhiên n lớn Explanation: While there isn't a simplification of (2n)! n!, there are other ways of expressing it. 3n >n2 3 n > n 2. Bài 4: Cấp số nhân. * Giả sử u k = 2k 3 − I must show that it converges to 2. Cho dãy There are n possible odd numbers upto 2n, so if all numbers upto 2n map to n different unique odd numbers, then if we have a set of n+1 numbers upto 2n, then atleast two of them map to the same odd number by pigeonhole principle. Take n elements and count how many ways there are to put these two elements into 2 different containers (A and B) a) Every element can take two states: it's either in A or in B.2n)1 + n n ( = nu . tức là luôn tồn tại n ≥ + 1 để 2 - 1 > M. e) Conjecturer la limite de la suite (un). Middle School Math Solutions - Inequalities Calculator. Bài 3: Cấp số cộng. Even more succinctly, the sum can be written as. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Câu hỏi Nhận biết. Số hạng thứ tám của dãy số là: A. 5. Given that Ug = 2u4^2, find the value of the constant k and prove that for all values of n, un ~ Un-1 = 4n + 3. 15, 2022 11:38 p. 2 of 2 | . Simplify (2n+2) (2n-2) (2n + 2) (2n − 2) ( 2 n + 2) ( 2 n - 2) Expand (2n+2)(2n− 2) ( 2 n + 2) ( 2 n - 2) using the FOIL Method.. Claim: an does not converge to − 1 2. From this point, I was stuck. + ta có: Chọn B..1. 4.2. Assuming the statement is true for n = k: 12 + 22 + 32 + + k2 = k(k + 1)(2k + 1) 6; (1) we will prove that the statement must be true for n = k + 1: Dãy số u n cho bởi u 1 = 3, u n + 1 = 1 + u n 2 , n > 1. Step by step video & image solution for For U_n=int_0^1x^n (2-x)^n dx ; V_n=int_0^1x^n (1-x)^ndxn in N , which of the following statement (s) is/are true? (a)U_n=2^n V_n (b) U_n=2^ (-n)V_n (c)U_n=2^ (2n)V_n (d) V_n=2^ (-2n)U_n by Maths experts to help you in doubts & scoring excellent marks in Class 12 exams. (2n)! n! = 2nn−1 ∏ k=0(2k +1) = 2n(1 $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1 Pour résoudre l'équation, factorisez n^{2}-2n-15 à l'aide de la n^{2}+\left(a+b\right)n+ab=\left(n+a\right)\left(n+b\right) de formule. In this video I prove by induction that 3^(2n + 1) + 2^(n + 2) is divisible by 7 for all nonnegative integers n. Claim: an does not converge to − 1 2. Exercice 3. 43872 views around the world You can reuse this answer Creative Commons License Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. I am using induction and I understand that when n = 1 n = 1 it is true.(2n-1)} 2 n. Bài 5: Ôn tập chương Dãy số. Add a comment. Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values. un = exp(cos(1 n)) − exp(cos(2 n)) 3.N. Dãy số un = 2n+ √n2 +4 u n = 2 n + n 2 + 4 có bao nhiêu số hạng làng số nguyên. I am using induction and I understand that when n = 1 n = 1 it is true. The sum on the left is 1/2 of the harmonic series, which diverges. UNITED NATIONS (AP) — The U. What is induction in calculus? In calculus, induction is a method of proving that a statement is true for all values of a variable within a certain range. Popular Problems . Hint: One very useful property of factorials is that. For each rational number x write x = p/q where p, q are integers with no common factors and q > 0, and then define f(x) = 1/q.1 4. en. So, would anyone please help me to Click here:point_up_2:to get an answer to your question :writing_hand:let u n dfrac n n2 where n in n Ex 9. d) À l’aide de la calculatrice, calculer u10, u100 et u1 000. The sum ∑(n(n + 1)(2n + 1))/4 for n [n=1, 7] is Then. C. Factor using the perfect Cách 2: Đánh giá trực tiếp. Barisan.3 n; (B) 9 n (C) 3 n + 3 (D) 6n.doc. Substitute n = k (this is different from the previous n) into the equation Un+2 − 3Un+1 + 2Un = 0: Therefore, the equation is true for n = k + 2. For example, the recurrence relation for the Fibonacci sequence is (This, together with the initial conditions and give the entire recursive definition for the sequence. 28, 2022, at United Nations Headquarters. Apr 8, 2013 at 3:45. Vậy dãy số (un) là dãy số giảm. We have ∞ ∑ n = m n! n!(n + 1)(n + 2)⋯2n ≤ ∞ ∑ n = m 1 n2 < ∞. an = n!/2^n Determine whether the sequence converges or diverges. Middle School Math Solutions – Inequalities Calculator. Số hạng u 2n bằng: (A) 2. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. mais le dénominateur commun que tu as ( 2 n+1 * 2 n) est lourd. D.sediuG . Identify the Sequence 4, 12 Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. P 1 n=1 nsin2 3+1 Answer: We know that jsinnj<1, so nsin2 n n3 + 1 n n3 + 1 n Trong đó, un = u(n) hoặc viết tắt là (un), và gọi u1 là số hạng đầu, un là số hạng thứ n và là số hạng tổng quát của dãy số.N. Demostración (parte 1 de 2): La suma de los n primeros números al cuadrado es n(n+1)(2n+1)/6 Demostración a través del método de inducción matemática complet 4. So for the induction step we have n = k + 1 n = k + 1 so 3k+1 > (k + 1)2 3 k + 1 > ( k + 1) 2 which is equal to 3 ⋅3k > k2 + 2k + 1 3 ⋅ 3 k > k 2 + 2 k + 1. However, there are some sequences for which it is hard to tell whether they are geometric or arithmetic (as the one described above), and yet, there are By. So our series is; S = ∞ ∑ n=0 n2 2n. Visit Stack Exchange 2. Do đó, dãy số (un) với un = 2n + 1 n + 2 bị chặn dưới.. Write QED or or // / / or something to indicate that you have completed your proof. Choose "Identify the Sequence" from the topic selector and click to see the result in our Algebra Calculator ! Examples . Ikut Bimbel online CoLearn mulai 95. C. Aug. However, activists and doctors' groups say the real toll is far higher. u n = 2 n + 1 n + 2. Tìm số hạng u17 u 17 . 12n2 +2n−4 = 0 ← ÷2 3n2+2n-1=0 Two solutions were found : n = -1 n = 1/3 = 0. To find the 1st term, put n = 1 into the formula, to find the 4th term, replace the n's by 4's: 4th term = 2 × 4 = 8. This is done by showing that the statement is true for the first term in the range, and then using the principle of mathematical induction to show that it is also true for all subsequent terms. Share. You can prove it for all real values n ≥ 2. The results of a draft resolution vote are seen on a screen as the UN general assembly holds an emergency special session on the Israel-Hamas war at the UN on 12 … 6. Prove that (2 n! Prove that:(2n)!/n! = { 1*3*5.S. 660 military observers and staff officers, and 2. The conflict in Sudan has wrecked the country and killed up to 9,000 people as of October, according to the United Nations. an + 1 + 1 2 = 2a2n − 1 + 1 2 = 2a2n − 1 2 = 2(an + 1 2)(an − 1 2) 4. Ví dụ 3: Cho dãy số (un ) được xác định bởi u 1 = 1 và với mọi n ≥ 2. UN votes unanimously to start the withdrawal of peacekeepers from Congo by year's end. 12 + 22 + 32 + + n2 = n(n+ 1)(2n+ 1) 6 Proof: For n = 1, the statement reduces to 12 = 1 2 3 6 and is obviously true.N. Correction Exercice 1.1A, it was 2/ǫ−1, but any bigger number would do, for example N = 2/ǫ. b) Dễ thấy u n > 0 với mọi n ε N * Mặt khác, vì n ≥ 1 nên n 2 ≥ 1 và 2n ≥ 2. Số hạng Un+1 bằng - Tuyển chọn giải bài tập Toán lớp 11 hay, ngắn nhất bám sát chương trình sách giáo khoa Toán 11 để giúp bạn học tốt môn học Toán 11. Figure 4. Bài 1: Phương pháp quy nạp toán học. How do I proceed from here? zhw. Prove that 2n < (2n n) < 22n.S. By EDITH M. ∑∞ n=1 nxn ∑ n = 1 ∞ n x n , or ∑∞ n=0 nxn ∑ n = 0 ∞ n x n. [Math Processing Error] u 8 = 63.2n−1 − 3 u Xét tính tăng, giảm và bị chặn của dãy số (un) u n, biết:un = 2n−13 3n−2 u n = 2 n - 13 3 n - 2. Diverges by the Limit Comparison Test Due to the simplicity of the series, we can use the Limit Comparison Test, which tells us if we have some positive sequence b_n and we know the convergence or divergence of sumb_n, then if c=lim_(n->oo)a_n/b_n>0 ne oo, then both series either converge or diverge. - Ví dụ: a) Dãy các số tự nhiên chẵn: 2; 4; 6; 8; …có số hạng đầu u1 = 2, số hạng tổng quát là un = 2n. The United States, … Solve your math problems using our free math solver with step-by-step solutions. General Assembly passes resolution calling for a ceasefire in Gaza 153 countries vote in favour of resolution, 23 abstain; U. D.e. + Số hạng thứ 10 là => C sai. The value of ∑(from n = 1 to N Un, if Un = |(n, 1, 5) (n^2, 2N+1, 2N+1) [0,∞] and z = ∑ cos^2n θ sin^2n θ for n ∈ [0,∞], 0 < θ < π/2, then show that xyz = x + y + z. Specifically, you can argue as follows. When we talk about algorithms, we often say that time complexity grows is exponential. The nth term of a sequence, Un, is given by Un = n(2n + k). Tìm số hạng thứ 4 a n. Cho dãy số (un) ( u n) có un = n2 +2n u n = n 2 + 2 n. Differentiation. a) Calculer les cinq premiers termes de la suite (un)., the further out you must go for the approximation to be valid within ǫ . Apply that to the product $$\frac{n!}{2^n}\: =\: \frac{4!}{2^4} \frac{5}2 \frac{6}2 \frac{7}2\: \cdots\:\frac{n}2$$ This is a prototypical example of a proof employing multiplicative telescopy. Step 1/5 First, let's find the value of the constant k.