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 Σa_k, we can introduce the corresponding series

$\sum _{k=1}^{\mathrm{\infty }}|a_k|=|a_1|+|a_2|+|a+3|+\dots$

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 $\sum _{k=1}^{\mathrm{\infty }}{a}_k$ is absolutely convergent if the series of absolute values $\sum _{k=1}^{\mathrm{\infty }}|a_k|$ is convergent.

A few simple examples demonstrate the concept of absolute convergence.

Example: Convergent p-Series

The infinite series

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

is absolutely convergent because

$\sum _{k=1}^{\mathrm{\infty }}\left|\frac{(-1{)}^k}{k^2}\right|=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2}$

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

Example: Harmonic Series

The infinite series

$\begin{array}{r}\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^k}{k}\end{array}$

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

$\sum _{k=1}^{\mathrm{\infty }}\left|\frac{(-1{)}^k}{k}\right|=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}$

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).