1. (Exercise 4.10, modified) Given a series Σ 1 ak with ak 0 for all k and lim Qk+1 k0oak we will prove that the series converges absolutely. (This is part of the ratio test sce the handout.) (a) Fix...
Determine whether the following series converges absolutely, converges conditionally, or diverges. 00 (-1)+1e 3k Σ-11: -Σ ak (k 17 k 1 k 1 Find lim a. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. koo O A. lim ak koo O B. The Ilimit does not exist. (1)* 1 (k 17) 3k e Σ. Now, let denote What can be concluded from this result using the Divergence Test? k 1 O...
Determine whether the following series converges. 0 Σ 8(-1) 2k + 5 k=0 Let ak 20 represent the magnitude of the terms of the given series. Select the correct choice below and fill in the answer box(es) to complete your choice. A. The series converges because ak = of k>N for which ak+1 Sak: and for any index N, there are some values of k>N for which ak+1 ? ak and some values B. The series converges because ak =...
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...
(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...
Use the Ratio Test to determine whether the series converges ab 00 2k Σ k 149 k= 1 Select the correct choice below and fill in the answer box to compl (Type an exact answer in simplified form.) O A. The series converges absolutely because r = OB. The series diverges because r= O c. The Ratio Test is inconclusive because r=
all part of one question
Determine whether the following series converges absolutely, converges conditionally, or diverges. OD (-1)"ax= k1 k=1 Vk 14 +9 Find lim ak. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. k-20 OA. lim ax - OB. The limit does not exist. (-1*45 Now, let a denote E What can be concluded from this result using the Divergence Test? 14 k=1 Vk +9 O A. The series Elak...
part e and f
0 for all k E N and Σ at oo. For each of the following, either prove that the given series con- 4. Suppose ak verges, or provide an example for which the series diverges. ak 1 + at ar ai ak
0 for all k E N and Σ at oo. For each of the following, either prove that the given series con- 4. Suppose ak verges, or provide an example for which the series...
(1 point) The series is an alternating series but we can apply the ratio test to to test for absolute convergence. Applying the ratio test for absolute convergence you would compute lim (k+1 = li k00 ak k- 00 Hence the series converges Note that you will have to simplify your answer for the limit or you will get an error message.
Prove the ratio test . What does this tell you if
exists?
(Ratio test) If
for all sufficiently large n and some
r < 1, then
converges absolutely; while if
for
all sufficiently large n, then
diverges.
lim |.1n+1/01 700 In+1/xn < We were unable to transcribe this image2x+1/2 > 1 We were unable to transcribe this image
2 (1) For z E C, define exp z - n-0 (a) Prove that the infinite series converges absolutely for z E C (b) Prove that when z є R, the definition of exp z given above is consistent with the one given in problem (2a), assignment 16. Definition from Problem (2a): L(x(1/t)dt E(z) = L-1 (z)
2 (1) For z E C, define exp z - n-0 (a) Prove that the infinite series converges absolutely for z E C...