1. Here is a sequence of partial sums of the series ak: 5n+3 n+4 / k=...
1 n+00 2 n=1 A sequence {$n} of partial sums of the series an has the property that lim Sn = Which of the following is true? 1 (a) lim an = 0. (b) lim an (c) lim an does not exist. (d) There is no way to determine the value of lim an. n+00 noo n+00 n+00 1 n The sequence {en} of partial sums of the series an has the property that sn = n=1 for every positive...
Consider the series a. List the nth term, Sn, of the sequence of partial sums for this series. b. What does the series converge to?
(Exercise 4.13, reordered) Given a series ΣΧί ak, let 8,-Ση-i ak. Σχί ak is Cesaro summable if S1 + 82 +... +Sn lim n-+o converges. (a) Give an example of a series Σ00i ak that is Cesaro sum mable but not convergent (b) Prove that if 1 ak converges, then it is Cèsaro summable. Hint: Say the sequence of partial sums sn → L. Try to prove that =1 8k → L by showing and then splitting the latter sum...
3.) Let ak E R with ak > 0 for all k E N. Suppose Σ㎞iak converges. Show that Σί1bk (By definition, for a sequence (ck), we say liCkoo if, for all M ER with Hint: Show that there exists (Ni))ไ1 with N > Nj for all j E N, such that bk there exists a sequence (bk)k of real numbers such that lim converges = oo and M >0, there exists N E N such that ck > M...
o Consider the series an whose partial sums are denoted as Sn for n > 1. n= 1 (1) If an = m+2, does an converge or diverge? Explain. n = 1 (2) If Sn = 7+2, does [ an converge or diverge? Explain. 111
3. Consider the sequence {ak} = 1 = 1. -1 1 -1 1 -1 2 5 6 Select co ܠܛ the true statement: A The sequence {ak} and the series Ļak both diverge. B. The sequence {ak} converges, but the series [ak diverges. c| The sequence {ak} diverges, but the series [ak converges. D| The sequence {ak} and the series ak both converge. E None of the above.
Please write it clearly and show every step ere Cesaro Sumrnability. Given an infinite series Σ an let Sn be the sequence of partial sums and let 5 Tt A series is Cesaro-surmable if linn-troƠn exists (and is finite). and this limit is called the Cesàro sum (a) Given the series 2n-1 n' s", hnd 8m and Ơn for any 1. and find the Cesaro sum of ΣΥ_1)". (b) Find the Cesàro sum of Here you may use the fact,...
Calculate the first eight terms of the sequence of partial sums correct to four decimal places. 1 6,0000 3 4 6 7 8 Does it appear that the series is convergent or divergent? convergent divergent Need Help? Read It Talk to a Tutor Calculate the first eight terms of the sequence of partial sums correct to four decimal places. 1 6,0000 3 4 6 7 8 Does it appear that the series is convergent or divergent? convergent divergent Need Help?...
problem 1and 2 Problem 1 [3 marks] Assume that the nth term in the sequence of partial sums for the series ,, is given below. Determine if the series is convergent or divergent. If the series is convergent determine the value of the series. a) Sn = 2-72 b) SEP Problem 2 [2 marks] Does the series (-1)" cos converge absolutely, or diverge?
2. Consider the sequence {2(-1)"}=1 (a) List the first 4 terms. (b) Compute for the partial sum of SA (e) Determine if the series converge or diverge. If it does converge what value it converges to. 00 2-3) nal