2. Prove that if an converges, then a also does.n=1n=13. Show that if an=1√anbn converges.n=1an and bn are two convergent series of non-negative numbers, thenn=14. Suppose that an diverges where an > 0. What can we conclude about the seriesn=1n=1an1+ nanand Σn=1an1+n²an

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Answered: Prove that if Σan converges, then Σa…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let {an}=1 and {bn}=1 be sequences. Prove or disprove: If {an – bn}=1 converges to 0 and {an}=1 converges to A, then {bn}-1 also converges to A.
- Let Σa, and Σb, are two infinite series such that 05a, bn, thes a) ifa, converges then Σb, converges. if diverges thea Σa, diverges. e) if Σb, converges then Σa, converges. d) if Σa, converges then Σb, diverges. Let (F,,.) be a field and f(x) be irreducible polynomial in F(x) then a) F(x) is a field. b) the principal ideal is the maximal ideal and need not to be a prime ideal. c) the principal ideal is a prime ideal and need not to be maximal ideal. d) F(x)/ is an integral domain but not a field.
PROPOSITION 3.2. A sequence (an) converges to a if and only if every subsequence of (an) also converges t a.
6. Prove the first part of FACT 2, i.e., if an converges and bn diverges, then Σ(an + bn) diverges. (Hint: do a proof by contradiction and invoke Theorem 8.)
If ) Jan diverges and a, converges, what is our conclusion? n=1 (a) An Converges absolutely. n=1 (b) Y An Converges conditionally. n=1 (c) an converges conditionally n=1 (d) an diverges. n=1
Suppose that ∑ an converges absolutely. Prove that ∑ an2 converges. If we replace "converges absolutely" with "converges conditionally" does this still remain true? If yes, give a proof, if no, find a counterexample.
,(-1)" converges conditionally, but not absolutely. Show all work. 3n Prove E
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3). True or False. If true prove it, if false give a counter example a) Jf Σax converges absolutely then Zak converges 5) If Σax converges then Zax converges absolutely Σακ
Let an 6n 2n + 1 (a) Determine whether {a} is convergent. O convergent divergent = (b) Determine whether Σ n = 1 convergent O divergent a, n is convergent.
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Prove that if Σan converges, then Σa also does.