Convergência Absoluta

A series is absolutely convergent if the series consisting of the absolute values of the terms is convergent. More formally, $\sum_{i=1}^\infty a_i$ is absolutely convergent iff $\sum_{i=1}^\infty |a_i|.$

Nice job! The intuition is that the condition for absolute convergence is "harder to satisfy," so if we know a series is absolutely convergent, instead of just convergent, we know more about the series.

The series $\sum a_n$ is said to be absolutely convergent if the series $\sum|a_n|$ is convergent.

Notice that absolute convergence implies convergence. Suppose $\sum a_n$ is absolutely convergent. Since (by the definition of absolute value) $$0\leq a_n + |a_n|\leq 2|a_n|,$$ by the series comparison test we know that $\sum a_n + |a_n|$ is convergent.

By definition of absolute convergence, $\sum |a_n|$ is convergent. Subtracting shows that $\sum (a_n+|a_n|) - |a_n| = \sum a_n$ is convergent, as desired.

Here we take the absolute value of each term, which eliminates the ugly term with the -1. Then since $\sum_{n=2}^{\infty}\frac{1}{n^3}$ converges, so does the given series.

It is worth noting that the converse does not hold: convergent series aren't necessarily absolutely convergent.