True or False If ∑k=1^∞ ak and ∑k=1^∞ bk are both series of positive terms and if limn →∞ (an)/(

Name: Problem 3, Chapter 8: Calculus, Early Transcendentals
Uploaded: 2021-08-02T03:15:39-08:00
Duration: 29 s
Channel: Adrian Co
Description: True or False If ∑k=1^∞ ak and ∑k=1^∞ bk are both series of positive terms and if limn →∞ (an)/(bn)=L, where L is a positive real number, then the series to be tested converges.

step 1 is asking us if we have A sub K, B, sub K, in both series of positive terms, if that the limit a

Question

True or False If $\sum_{k=1}^{\infty} a_{k}$ and $\sum_{k=1}^{\infty} b_{k}$ are both series of positive terms and if $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=L,$ where $L$ is a positive real number, then the series to be tested converges.

   True or False If $\sum_{k=1}^{\infty} a_{k}$ and $\sum_{k=1}^{\infty} b_{k}$ are both series of positive terms and if $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=L,$ where $L$ is a positive real number, then the series to be tested converges.

Calculus, Early Transcendentals

Michael Sullivan,… 1st Edition

Chapter 8, Problem 3

Instant Answer

Solved by Expert Adrian Co

08/02/2021

Step 1

We are also given that $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=L,$ where $L$ is a positive real number. Show more…

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True or False If $\sum_{k=1}^{\infty} a_{k}$ and $\sum_{k=1}^{\infty} b_{k}$ are both series of positive terms and if $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=L,$ where $L$ is a positive real number, then the series to be tested converges.

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Key Concepts

Limit Comparison Test

The limit comparison test is a tool used in the analysis of infinite series with positive terms. It states that if two series have terms a? and b? (with all terms positive) and the limit of a?/b? as n approaches infinity is a finite number L > 0, then both series will either converge or diverge together. This test allows one to infer the convergence behavior of a series by comparing it to another series whose convergence is already known.

Convergence of Series with Positive Terms

The convergence analysis of series with positive terms relies on methods that exploit the non-negativity of the series terms. Since the partial sums of such series are monotonic, various comparison tests (including the limit comparison test) can be effectively used to determine whether the series converges or diverges. These techniques form a fundamental part of understanding infinite sums in mathematical analysis.

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