# Alternating Series Test

## Theorem

Let $\sequence {a_n}_{N \mathop \ge 0}$ be a decreasing sequence of positive terms in $\R$ which converges with a limit of zero.

That is, let $\forall n \in \N: a_n \ge 0, a_{n + 1} \le a_n, a_n \to 0$ as $n \to \infty$

Then the series:

$\ds \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} a_n = a_1 - a_2 + a_3 - a_4 + \dotsb$

## Proof

First we show that for each $n > m$, we have $0 \le a_{m + 1} - a_{m + 2} + a_{m + 3} - \dotsb \pm a_n \le a_{m + 1}$.

### Lemma

For all natural numbers $n, m$ with $n > m$ we have:

$\ds \sum_{k \mathop = m + 1}^n \paren {-1}^k a_k \le a_{m + 1}$

$\Box$

Therefore for each $n > m$, we have:

$0 \le a_{m + 1} - a_{m + 2} + a_{m + 3} - \dotsb \pm a_n \le a_{m + 1}$

Now, let $\sequence {s_n}$ be the sequence of partial sums of the series: :$\ds \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} a_n$

Let $\epsilon > 0$.

Since $a_n \to 0$ as $n \to \infty$:

$\exists N: \forall n > N: a_n < \epsilon$

But $\forall n > m > N$, we have:

 $\ds \sequence {s_n - s_m}$ $=$ $\ds \size {\paren {a_1 - a_2 + a_3 - \dotsb \pm a_n} - \paren {a_1 - a_2 + a_3 - \dotsb \pm a_m} }$ $\ds$ $=$ $\ds \size {\paren {a_{m + 1} - a_{m + 2} + a_{m + 3} - \dotsc \pm a_n} }$ $\ds$ $\le$ $\ds a_{m + 1}$ from the above $\ds$ $<$ $\ds \epsilon$ as $m + 1 > N$

Thus we have shown that $\sequence {s_n}$ is a Cauchy sequence.

The result follows from Convergent Sequence is Cauchy Sequence.

$\blacksquare$

## Also known as

The Alternating Series Test is also seen referred to as Leibniz's Alternating Series Test.

## Historical Note

The Alternating Series Test is attributed to Gottfried Wilhelm von Leibniz.