Question

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.

Short answer

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

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.

$$\begin{gathered} \sum _{k=1}^{\infty }{a}_k=\sum _{k=1}^{\infty }\frac{1}{k} \\ =\underset{k\to \infty }{lim}\frac{1}{k} \\ =0 \end{gathered}$$

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