Alternating Series Test

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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:

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

converges.


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:

\(\ds \sequence {s_n - s_m}\) \(=\) \(\ds \size {\paren {a_1 - a_2 + a_3 - \dotsb \pm a_n} - \paren {a_1 - a_2 + a_3 - \dotsb \pm a_m} }\)
\(\ds \) \(=\) \(\ds \size {\paren {a_{m + 1} - a_{m + 2} + a_{m + 3} - \dotsb \pm a_n} }\)
\(\ds \) \(\le\) \(\ds a_{m + 1}\) 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, for Gottfried Wilhelm von Leibniz.

Some sources hyphenate: Alternating-Series Test


Historical Note

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


Sources