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:
\(\ds \sequence {s_n - s_m}\) | \(=\) | \(\ds \size {\paren {a_1 - a_2 + a_3 - \dotsb \pm a_n} - \paren {a_1 - a_2 + a_3 - \dotsb \pm a_m} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \size {\paren {a_{m + 1} - a_{m + 2} + a_{m + 3} - \dotsb \pm a_n} }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds a_{m + 1}\) | 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
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 6.13$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): alternating series test or Leibniz's alternating series test
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests: Theorem $1.5$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): alternating series
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): alternating series
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): alternating series test