Question

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition or description with a graph or an algebraic example.

An alternating series

Short answer

Let $a_n$be a positive sequence. An alternating series is the series of either the form $\sum _{n=1}^{\infty }{\left(-1\right)}^n{a}_{n}or\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{a}_n$

For example,$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{1}{n}$is an alternating series.

Step-by-step solution

Step 1. Given Information    

The given statement is an alternating series

Step 2. Explanation     

Let $a_n$be a positive sequence. An alternating series is the series of either the form role="math" localid="1649153725164" $\sum _{n=1}^{\infty }{\left(-1\right)}^n{a}_{n}or\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{a}_n$For example,$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{1}{n}$is an alternating series.