Show that if ak is absolutely convergent and |bx| < |ak| for all k, then E bk is absolutely convergent. (a) Show that if E ak is absolutely convergent, then Ea? is convergent. (b) Show by means of an example that the converse of the result in part (a) is false.

Q: Assume that En=1 an is convergent, and denote by b = a3k-2 + a3k-1 + a3k for k > 1. (a) Prove that…

A: This is a problem of sequence and series.

Q: An. If > an converges for some large k, then so will n=1 n=k Select one: True False

A:

Q: 1. Use an appropriate way to determine whether the following is convergent or divergent. k 'Ink i. Σ…

A:

Q: (2!)2 C)) (2n)! n=1

A:

Q: Suppose an is conditionally convergent. Define what this means, and then use a proof by n=1…

A:

Q: Rigorously show that (-1)"-1. n2 +1 is conditionally convergent. n=1

A:

Q: 1 Prove that if a sequence {a„} converges to a, then the sequence {[an]} converges to |a|. Find…

A: Suppose {an} converges to a. Prove that an converges to a. Also, check the converse is true.

Q: 1 Determine whether (-1)". converges conditionally, absolutely, or not at all. Vn2 + 1 n=0

A:

Q: Supposean converges conditionally (but not absolutely). Let A+ = {n €N : an ≥ 0} n=0 and A_ = {n € N…

A:

Q: (Bolzano-Weierstrass Property) Does every sequence of points in E have a subsequence that converges…

A: Consider the sequence

Q: 5. Suppose > an is absolutely convergent. Prove > bn is absolutely convergent if |b,| 1.

A:

Q: (a) Prove that if > ak converges absolutely and (bk) is a subsequence of (ar) then ) bk also k=1 k=1…

A:

Q: 2n Let an = 3n +5 Determine whether {an} is convergent. Determine whether an is convergent.

A: To determine whether the given sequence and series is convergent or not.

Q: 3. Determine if the sequence xn = 47² is convergent or not and find its limit if it exists n²+1

A: We want to find the given sequence is convergent or divergent.

Q: Determine if >n=1 (-1)" is absolutely convergent or conditionally divergent. 4n 4 +1

A: First check for Absolutely convergence using Series Integral Test

Q: Suppose (an) is Cauchy and that for every k ∈ N, the interval (−1/k, 1/k) contains at least one term…

A: Given, an is Cauchy sequence, Let, ank∈-1k,1k ,∀k∈ℕ Then, ank1k , ∀k∈ℕ Let, ε&gt;0 be arbitrary and…

Q: 3 - If E is convergent, find its sum. z=1 en n(n+3)

A:

Q: 5. Suppose a, is absolutely convergent. Prove b, is absolutely convergent if |b,| 1.

A: This question is related to series comparison test.

Q: 1 3. En=1 (2n+1)3

A:

Q: Which of the followings are true? 1.En 1 nInn (-1)" is absolutely convergent. 11. En 1 (-1) is…

A: Option (e) is correct.

Q: Determine whether E=1(-1)"+1. 2n converges or diverges. If it converges, find the sum. 3n

A:

Q: d) Determine whether ) (an + bn) is convergent or divergent. n=1

A:

Q: (2) If £ an and D bn are both divergent then is E an · bn necessarily divergent? If yes, provide a…

A:

Q: e-n n(n + 1) E(-1)"+1. n=l

A:

Q: If {fn}n=1 is a pointwise convergent, then {fn}=1 is uniformly convergent.

A:

Q: {*} An If sequences {an} and {bn }both diverges to infinity, then the sequence is bn convergent.

A: Given query is to find the statement is true or false.

Q: 6 Let an 3n + 1 (a) Determine whether {a,} is convergent. convergent O divergent (b) Determine…

A: We can find the answer as below.

Q: If an and bn both converge, then an bn converges. n=1 n=1 n=1 True False

A:

Q: Let a < b be real numbers and assume a sequence (Tk)ž is strictly decreasing and Xk E (a, b) for…

A:

Q: (k!)* (2k)! k=1

A:

Q: If E Un is absolutely convergent then (un)² is also n=1 n=1 convergent.

A:

Q: Let a (> 1) be an irrational number. Show that the kth convergent of the simple continued fraction…

A:

Q: 5. If En-1 an converges, and if an +1 and a, > 0 for all n, (a) Show that E a converges. (b) Does E1…

A: Since, the sequence is a positive monotonically decreasing sequence, so, is also monotonically…

Q: ,(-1)" converges conditionally, but not absolutely. Show all work. 3n Prove E

A:

Q: 1 diverges if it is true. An 6. Suppose An converges. Assume the terms an are non-zero. Prove n=1…

A:

Q: Suppose an is conditionally convergent. Define what this means, and then use a proof by n=1…

A:

Q: Determine if the given is convergent or divergent. I In"(n*) 23n2 n=1

A:

Q: Let an 4n 7n + 1 (a) Determine whether {a} is convergent. convergent divergent = 8 (b) Determine…

A:

Q: 2. Let 0 b, for each n e N. With the fact in part (a), by simple computations (you should try it as…

A:

Q: EXAMPLE f(z) = Σo(z"/n!) is convergent for all z and, according to Theorem 2.9, n7"-1 8 -n

A: Here we have to show step wise solve.

Q: Suppose that the sequence {an} is monotone. Prove that {an} converges if and only if {a%} converges.

A:

Q: 7) : Determine if absolutely convergent, conditionally Question # convergent, or divergent.…

A: We use definition of a absolutly convergent series

Q: n· 3n 5n n=

A:

Q: (a) if Ek až is absolutely convergent then E ar is absolutely convergent.

A:

Q: diverges if it is true. an n=1 Suppose > an converges. Assume the terms an are non-zero. Prove > n=1…

A: Consider the given information: Let ∑n=1∞an is convergent series. The terms of an are non-zero. To…

Q: 4. Determine whether the scrics is absolutely convergent, conditionally convergent, or divergent. k…

A: Here we have, ∑k=1∞-1kkk3+2=∑k=1∞-1kak

Q: Determine whether {an} is convergent

A: First find the limit of the sequence &lt; an &gt;:

Q: 1. Prove that the sequence diverges but it has a convergent subsequence: (-1)" 2+(-1)", (a) (b) X =…

A:

Q: 6. Let E CR and E # 0. Prove if E is compact then every sequence in E has a subsequence that…

A: