Chapter 6: Chapter 6

Problem 4

Let \(a_{n}>0\), and suppose \(\sum_{n=1}^{\infty} a_{n}\) converges. If \(r_{n}=\sum_{k=n}^{\infty} a_{k}\), show that (a) \(\sum_{n=1}^{\infty} \frac{a_{n}}{r_{n}}\) diverges (b) \(\sum_{n=1}^{\infty} \frac{a_{n}}{\sqrt{r_{n}}}\) converges.

Short Answer

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(a) The series \(\sum_{n=1}^{\infty} \frac{a_{n}}{r_{n}}\) diverges according to the limit comparison test. (b) The series \(\sum_{n=1}^{\infty} \frac{a_{n}}{\sqrt{r_{n}}}\) converges also according to the limit comparison test.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Analyzing the provided series

Given \(a_{n}>0\) and the series \(\sum_{n=1}^{\infty} a_{n}\) converges, which implies the terms \(a_{n}\) go to zero as \(n\to\infty\). We also have \(r_{n}=\sum_{k=n}^{\infty} a_{k}\), known as the 'tail' of the sequence starting at \(n\), and clearly \(r_{n} \geq a_{n}\) for all \(n\), since each \(r_{n}\) is a sum of terms, all of which are positive.

Step 2: Application of the limit comparison test for (a)

We want to show \(\sum_{n=1}^{\infty} \frac{a_{n}}{r_{n}}\) diverges. Consider an easier series \( \sum_{n=1}^{\infty} \frac{a_{n}}{a_{n}} = \sum_{n=1}^{\infty} 1 \) which diverges. We have the ratio of corresponding terms of the two series as \(L=\lim_{n\to\infty} \frac{a_{n}/r_{n}}{a_{n}/a_{n}} = \lim_{n\to\infty} \frac{a_n}{a_n}.\frac{a_n}{r_n} = \lim_{n\to\infty} \frac{a_n}{r_n}\). If this limit is positive and finite, by the limit comparison test, then the series \(\sum_{n=1}^{\infty} \frac{a_{n}}{r_{n}}\) also diverges. But this limit is equal to 1 since \( a_n < r_n \) for all \(n\). Therefore by limit comparison test, the series \(\sum_{n=1}^{\infty} \frac{a_{n}}{r_{n}}\) diverges.

Step 3: Application of the limit comparison test for (b)

We want to show \(\sum_{n=1}^{\infty} \frac{a_{n}}{\sqrt{r_{n}}}\) converges. Consider an easier series \( \sum_{n=1}^{\infty} \frac{a_{n}}{a_{n}^{1/2}} = \sum_{n=1}^{\infty} a_{n}^{1/2} \) which converges (since \(\sum_{n=1}^{\infty} a_{n}\) converges and \(a_{n}\) go to zero as \(n\to\infty\)). Similar to the second step, we have the ratio of corresponding terms of the two series as \(L=\lim_{n\to\infty} \frac{a_{n}/\sqrt{r_{n}} }{a_{n}^{1/2} } =\lim_{n\to\infty} \frac{a_{n}^{1/2}.\sqrt{a_n}}{\sqrt{r_{n}} }=\lim_{n\to\infty} \frac{a_{n}}{\sqrt{r_{n}}} \). If this limit is positive and finite, by the limit comparison test, then both series converge or both diverge. But this limit is equal to 1 since \( a_n < r_n \) for all \(n\). Therefore by the limit comparison test, the series \(\sum_{n=1}^{\infty} \frac{a_{n}}{\sqrt{r_{n}}}\) converges.

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Chapter 6: Chapter 6

Let \(a_{n}>0\), and suppose \(\sum_{n=1}^{\infty} a_{n}\) converges. If \(r_{n}=\sum_{k=n}^{\infty} a_{k}\), show that (a) \(\sum_{n=1}^{\infty} \frac{a_{n}}{r_{n}}\) diverges (b) \(\sum_{n=1}^{\infty} \frac{a_{n}}{\sqrt{r_{n}}}\) converges.

Short Answer

Step by step solution

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Step 1: Analyzing the provided series

Step 2: Application of the limit comparison test for (a)

Step 3: Application of the limit comparison test for (b)

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More chapters from the book ‘Complex Variables With Applications’

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

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