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The goal of this problem is to justify the following statement by using the Comparison Test for series. Suppose that an and bn are convergent series with a,, bn 2 0 for all natural numbers n. Then, anb, is also a convergent series. (a) By only using the Comparison Test and the following fact (FACT 1), justify that if a, 20 for all n and a, is convergent, then a is also convergent.

The goal of this problem is to justify the following statement by using the Comparison Test for series. Suppose that an and bn are convergent series with a,, bn 2 0 for all natural numbers n. Then, anb, is also a convergent series. (a) By only using the Comparison Test and the following fact (FACT 1), justify that if a, 20 for all n and a, is convergent, then a is also convergent.

Advanced Engineering Mathematics
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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4. The goal of this problem is to justify the following statement by using the Comparison Test for series. Suppose that an and b, are convergent series with an, bn 20 for all natural numbers n=1 n=1 n. Then, anbn is also a convergent series. n=1 (a) By only using the Comparison Test and the following fact (FACT 1), justify that if a, 20 for all n and a, is convergent, then a is also convergent. n=1 n=1 2 FACT 1: if a, > 0 for all n and > an is convergent, then a < an for n > M for some n=1 natural number M.
Transcribed Image Text:4. The goal of this problem is to justify the following statement by using the Comparison Test for series. Suppose that an and b, are convergent series with an, bn 20 for all natural numbers n=1 n=1 n. Then, anbn is also a convergent series. n=1 (a) By only using the Comparison Test and the following fact (FACT 1), justify that if a, 20 for all n and a, is convergent, then a is also convergent. n=1 n=1 2 FACT 1: if a, > 0 for all n and > an is convergent, then a < an for n > M for some n=1 natural number M.
(b) By only using the Comparison Test and the following fact (FACT 2), justify that if an, bn >0 for all n and an, brn are convergent, then en is also convergent, n=1 n=1 n=1 where cn is the maximum of an and bn. FACT 2: If an, b, 2 0 for all n and En, Ebn are convergent, then an + b, is n=1 n=1 n=1 also convergent. (c) By only using the Comparison Test and statements in (a) and (b), justify that if an and bn are convergent and an, bn 2 0 for all natural numbers n, then n=1 n=1 anbn is also convergent. n=1
Transcribed Image Text:(b) By only using the Comparison Test and the following fact (FACT 2), justify that if an, bn >0 for all n and an, brn are convergent, then en is also convergent, n=1 n=1 n=1 where cn is the maximum of an and bn. FACT 2: If an, b, 2 0 for all n and En, Ebn are convergent, then an + b, is n=1 n=1 n=1 also convergent. (c) By only using the Comparison Test and statements in (a) and (b), justify that if an and bn are convergent and an, bn 2 0 for all natural numbers n, then n=1 n=1 anbn is also convergent. n=1
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