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.

Absolute convergence for a series

Short answer

A series is called absolutely convergent if $\sum _n\left|a_n\right|$ is convergent. For example,$\sum _{n=1}^{\infty }\frac{\sin n}{n^3}$

Step-by-step solution

Step 1. Given Information     

The given statement is Absolute convergence for a series

Step 2. Explanation      

A series is called absolutely convergent if $\sum _n\left|a_n\right|$is convergent.

Absolute convergence means a series will converge even when you take the absolute value of each term. If a series converges absolutely, it converges even if the series is not alternating. For example,$\sum _{n=1}^{\infty }\frac{\sin n}{n^3}$

In other words, a series converges absolutely if it converges when you remove the alternating part.