# Alternating Series Test

Most of the Convergence tests require the terms of an infinite series to be positive. At times we may have to deal with series whose terms are not necessarily positive.

In particular, it is important to learn the methods for testing the convergence of Alternating Series, the series whose terms alternate back and forth from positive and negative. In other words, an Alternating Series is a series whose terms are alternatively positive and negative.**Mathematically, an alternating series can be either expressed using Sigma notation or written in expanded form.**

$\sum_{n=1}^{\infty }(-1)^{n+1}a_{n}$ = $a_1 - a_2 + a_3 - a_4 + .......... + (-1)^{n-1} a_{n} + ............ ∞**Example:**

The series $\sum_{n=1}^{\infty }(-1)^{^{n+1}}$ $\frac{1}{n}$ = 1 - $\frac{1}{2}$ + $\frac{1}{3}$ - $\frac{1}{4}$ +........... is the Alternating Harmonic Series. It is interesting to note that the Alternating Harmonic Series is convergent, even though the General Harmonic Series 1 + $\frac{1}{2}$ + $\frac{1}{3}$ + $\frac{1}{4}$ + ............ is known to be a divergent series.