# Science:Infinite Series Module/Appendices/Proof of The Ratio Test/Absolute Convergence Implies Convergence

The ratio test uses the following theorem.

Theorem: Absolute Convergence Implies Convergence
If the infinite series

$\sum _{k=1}^{\infty }a_{k}$ is absolutely convergent, then it is convergent.

## Proof

By assumption,

$\sum _{k=1}^{\infty }a_{k}$ is absolutely convergent, meaning that

$\sum _{k=1}^{\infty }|a_{k}|$ is convergent. Therefore,

$\sum _{k=1}^{\infty }2|a_{k}|$ is also convergent. Since

$0\leq a_{k}+|a_{k}|\leq 2|a_{k}|$ then

$\sum _{k=1}^{\infty }a_{k}+|a_{k}|$ is convergent by the comparison test (the comparison can be found in most introductory calculus books that cover infinite series).

Finally,

$\sum _{k=1}^{\infty }a_{k}=\sum _{k=1}^{\infty }{\big (}a_{k}+|a_{k}|{\big )}-\sum _{k=1}^{\infty }=|a_{k}|$ must also be convergent because it is the difference of two convergent series.

## So What?

Simply put, if we can show that an infinite series is absolutely convergent, then we know that it is convergent. Sometimes, it's easier to show that a series is absolutely convergent than it is to show that it is convergent.