Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
for series. 8. Suppose that > an and bn are convergent series with an, bn > 0 for all natural numbers n=1 n=1 8. 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 an 20 for all n and > an is convergent, then > a, is also convergent. n=1 n=1

for series. 8. Suppose that > an and bn are convergent series with an, bn > 0 for all natural numbers n=1 n=1 8. 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 an 20 for all n and > an is convergent, then > a, is also convergent. n=1 n=1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 49E
See similar textbooks
icon
Related questions
Question
4. 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 an, bn > 0 for all natural numbers n=1 n=1 n. Then, >anbn is also a convergent series. n=1 By only using the Comparison Test and the following fact (FACT 1), justify that if an >0 for all n and >an is convergent, then > a, is also convergent. n=1 n=1 FACT 1: if an > 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 )`, > bn are convergent, then An, Cn is also convergent, n=1 where cn is the maximum of an and bn. n=1 n=1 FACT 2: If an, bn 2 0 for all n and An; bn are convergent, then An + bn is n=1 n=1 also convergent. n=1 (c) By only using the Comparison Test and statements in (a) and (b), justify that if > an and > bn are convergent and an, bn > 0 for all natural numbers n, then n=1 n=1 > anbn is also convergent. n=1
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 > bn are convergent series with an, bn > 0 for all natural numbers n=1 n=1 n. Then, >anbn is also a convergent series. n=1 By only using the Comparison Test and the following fact (FACT 1), justify that if an >0 for all n and >an is convergent, then > a, is also convergent. n=1 n=1 FACT 1: if an > 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 )`, > bn are convergent, then An, Cn is also convergent, n=1 where cn is the maximum of an and bn. n=1 n=1 FACT 2: If an, bn 2 0 for all n and An; bn are convergent, then An + bn is n=1 n=1 also convergent. n=1 (c) By only using the Comparison Test and statements in (a) and (b), justify that if > an and > bn are convergent and an, bn > 0 for all natural numbers n, then n=1 n=1 > anbn is also convergent. n=1
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
SEE MORE TEXTBOOKS