Alternating Series

A step-by-step guide to alternating series

An alternating series is a series in which the terms alternate between positive and negative. The general form of an alternating series is as follows:

\(\color{blue}{\sum \:\left(-1\right)^ka_{k}}\)

Where \(a_{k}\ge 0\) and the first index is arbitrary. It means that the starting term for an alternating series can have any sign.

We can say that an alternating series \([a_k]^\infty_{k=1}\) converges if two conditions exist:

• \(0\le a_{k+1}\le a_k\), for all \(k\ge 1\)
• \(a_k→0\), as \(k→+∞\)