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:

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


Sources

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