Suggested languages for you:

Americas

Europe

Q. 2

Expert-verified
Found in: Page 624

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# 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 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

## 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.

$\sum _{k=1}^{\infty }{a}_{k}=\sum _{k=1}^{\infty }\frac{1}{k}\phantom{\rule{0ex}{0ex}}=\underset{k\to \infty }{lim}\frac{1}{k}\phantom{\rule{0ex}{0ex}}=0$

## 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 $\sum _{k=1}^{\infty }{a}_{k}=\sum _{k=1}^{\infty }\frac{1}{{k}^{2}}$.