# 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 Σ*a _{k}*, we can introduce the corresponding series

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
is absolutely convergent if the series of absolute values
is convergent. |

A few simple examples demonstrate the concept of absolute convergence.

### Example: Convergent p-Series

The infinite series

is *absolutely* convergent because

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

### Example: Harmonic Series

The infinite series

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

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).