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Q. 2

Expert-verified

Calculus

Found in: Page 624

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) A divergent series $\sum_{k = 1}^{\infty} a_{k}$ in which $a_{k} \to 0$ .
(b) A divergent p-series.
(c) A convergent p-series.

(a) The example of the series is $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k}$ .

(b) The example of the series is $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k}$ .

(c) The example of the series is $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k^{2}}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Part (a) Step 1. Given Information.

A divergent series:

$\sum_{k = 1}^{\infty} a_{k}$

And $a_{k} \to 0$

Part (a) Step 2. Consider the given series.

Consider the given series.

$\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k} = \underset{k \to \infty}{l i m} \frac{1}{k} = 0$

Part (a) Step 3. Find the series.

So by using the harmonic series and p-test series, the series $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k}$ is divergent.

Part (b) Step 1. Find an example.

Consider the series,

$\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k}$

which is a harmonic series, and by p-series test, the series is divergent.

Part (c) Step 1. Consider the series.

Consider the series,

$\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k^{2}}$

which is convergent since $p = 2 > 1$ .

So the convergent p-series test is $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k^{2}}$ .

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