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

The ratio test requires the idea of absolute convergence. Given any infinite series Σak, we can introduce the corresponding series

${\displaystyle \sum _{k=1}^{\infty }|a_{k}|=|a_{1}|+|a_{2}|+|a+3|+\ldots }$

whose terms are the absolute values of the original series. We can explore whether this corresponding series converges, leading us to the following definition.

Definition: Absolute Convergence
The infinite series

${\displaystyle \sum _{k=1}^{\infty }a_{k}}$

is absolutely convergent if the series of absolute values

${\displaystyle \sum _{k=1}^{\infty }|a_{k}|}$

is convergent.

A few simple examples demonstrate the concept of absolute convergence.

### Example: Convergent p-Series

The infinite series

${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k^{2}}}}$

is absolutely convergent because

${\displaystyle \sum _{k=1}^{\infty }{\Bigg |}{\frac {(-1)^{k}}{k^{2}}}{\Bigg |}=\sum _{k=1}^{\infty }{\frac {1}{k^{2}}}}$

is a convergent p-series (p =2).

### Example: Harmonic Series

The infinite series

{\displaystyle {\begin{aligned}\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}\end{aligned}}}

is convergent (by the alternating series test), but is not absolutely convergent because

${\displaystyle \sum _{k=1}^{\infty }{\Bigg |}{\frac {(-1)^{k}}{k}}{\Bigg |}=\sum _{k=1}^{\infty }{\frac {1}{k}}}$

is the infamous harmonic series, which is not a convergent series.

It is possible for a series to be convergent, but not absolutely convergent (such series are termed conditionally convergent, but we do not need this definition for our purposes).