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: Suppose Σ XK is conditionally convergent, and define k=1 ak = max {xk, 0} and bk = min {xk, 0} ∞…

A: The objective of this question is to prove that the series Σa_k from k=1 to ∞ and Σb_k from k=1 to ∞…

Q: For each nЄ Z+ let fn : R → R be the function пх fn(2) n|x| +1° (a) Prove that (fn) converges…

A: Given the sequence of functions  defined by .(a)Consider the limit,This is true for every 0≠x∈R.…

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

A:

Q: 1. (a) Let an → 2. Prove from first principles (i.e. give a direct e-N proof) that a → 4. (b) Let a?…

A:

Q: b) Suppose (xn) does not converge to 0. Then there exists e > 0 such that all except for finitely…

A: Let us suppose the following sequence xn=1,0,1,0,1,0,..... We have to see that the odd terms are 1…

Q: Match each of the following with the correct statement. C stands for Convergent, D stands for…

A: Concept used: Divergence: When the values goes to -∞ or goes to ∞ that is it does not approach one…

Q: (-1)* k4/3 k=1 a) Converges both absolutely and conditionally b) Converges conditionally c)…

A:

Q: For eksample Xn = n² (n+1)! investigate whether the serles Σ 8 n=1 Xn convergent Or divergent. Prove…

A: We will use ratio test to check whether given series is convergent or not.

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

A:

Q: Suppose (xn) does not converge to 0. Then there exists e>0 such that all except for finitely many…

A: Consider the sequence xn=1,0,1,0,1,0,....... Here the odd terms are 1 and the even terms are 0.…

Q: 1).  State whether the statement is true, or give an example to show that it is false. If ∞     n =…

A:

Q: 1. Suppose for all n, an and bn are positive. If lim n-x Σan = 6 then Σαπ n=1 n=1 bn is also…

A: To solve the following problem

Q: (a) Directly use the Bolzano-Weierstrass theorem (Theorem 2.3.8) to prove that every Cauchy sequence…

A: Details attached in the next step.

Q: The n-th term in a sequence on is given by an - 6-e-n. & Is this sequence increasing, decreasing or…

A: MCT (Monotone Convergence Theorem ) :- A sequnce ,which is monotonic increasing and bounded above…

Q: 2.65 Let fn(x) = xe¬nx² a) Find || fn||sup for all n. b) Determine whether or not fn converges…

A: The given function is fn(x)=xe-nx2 ∀x∈ℝ To find: (a) ∥fn(x)∥Sup ∀n             (b) Whether…

Q: 80 7" (n!) 21 2" (2n +3)! n-1

A: Given series∑n=1∞(-1)n7n(n!)22n(2n+3)!