Alternating series test

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In mathematical analysis, the alternating series test is a method used to prove that an alternating series with terms that decrease in absolute value is a convergent series. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test or the Leibniz criterion.

Formulation[edit]

A series of the form

 \sum_{n=1}^\infty (-1)^{n-1} a_n = a_1 - a_2 + a_3 - \cdots \!

Or,

 \sum_{n=1}^\infty (-1)^{n} a_n = - a_1 + a_2 - a_3 + \cdots \!

where an are positive, is called an alternating series.

The alternating series test then says if {an} decreases monotonically and goes to 0 in the limit then the alternating series converges.

Moreover, let L denote the sum of the series, then the partial sum

S_k = \sum_{n=1}^k (-1)^{n-1} a_n\!

approximates L with error bounded by the next omitted term:

\left | S_k - L \right \vert \le \left | S_k - S_{k+1} \right \vert = a_{k+1}.\!

Proof[edit]

Suppose we are given a series of the form \sum_{n=1}^\infty (-1)^{n-1} a_n\!, where  \lim_{n\rightarrow\infty}a_{n}=0 and  a_n \geq a_{n+1} for all natural numbers n. (The case \sum_{n=1}^\infty (-1)^{n} a_n\! follows by taking the negative.) [1]

Proof of convergence[edit]

We will prove that both the partial sums S_{2m+1}=\sum_{n=1}^{2m+1} (-1)^{n-1} a_n with odd number of terms, and S_{2m}=\sum_{n=1}^{2m} (-1)^{n-1} a_n with even number of terms, converge to the same number L. Thus the usual partial sum S_k=\sum_{n=1}^k (-1)^{n-1} a_n also converges to L.

The odd partial sums decrease monotonically:

 S_{2(m+1)+1}=S_{2m+1}-a_{2m+2}+a_{2m+3} \leq S_{2m+1}

while the even partial sums increase monotonically:

 S_{2(m+1)}=S_{2m}+a_{2m+1}-a_{2m+2} \geq S_{2m}

both because an decrease monotonically with n.

Moreover, since an are positive,  S_{2m+1}-S_{2m}=a_{2m+1} \geq 0 . Thus we can collect these facts to form the following suggestive inequality:

 a_1 - a_2 = S_2 \leq S_{2m} < S_{2m+1} \leq S_1 = a_1.

Now, note that a1a2 is a lower bound of the monotonically decreasing sequence S2m+1, the monotone convergence theorem then implies that this sequence converges as m approaches infinity. Similarly, the sequence of even partial sum converges too.

Finally, they must converge to the same number because

 \lim_{n\to\infty}S_{2m+1}-S_{2m}=\lim_{n\to\infty}a_{2m+1}=0.

Call the limit L, then the monotone convergence theorem also tells us an extra information that

 S_{2m} \leq L \leq S_{2m+1}

for any m. This means the partial sums of an alternating series also "alternates" above and below the final limit. More precisely, when there are odd (even) number of terms, i.e. the last term is a plus (minus) term, then the partial sum is above (below) the final limit.

This understanding leads immediately to an error bound of partial sums, shown below.

Proof of partial sum error bound[edit]

We would like to show \left| S_k - L \right| \leq a_{k+1}\! by splitting into two cases.

When k = 2m+1, i.e. odd, then

\left| S_{2m+1} - L \right| = S_{2m+1} - L \leq S_{2m+1} - S_{2m+2} = a_{(2m+1)+1}

When k = 2m, i.e. even, then

\left| S_{2m} - L \right| = L - S_{2m} \leq S_{2m+1} - S_{2m} = a_{2m+1}

as desired.

Both cases rely essentially on the last inequality derived in the previous proof.

For an alternative proof using Cauchy's convergence test, see Alternating series.

For a generalization, see Dirichlet's test.

Notes[edit]

  1. ^ The proof follows the idea given by James Stewart (2012) “Calculus: Early Transcendentals, Seventh Edition” pp. 727–730. ISBN 0-538-49790-4

See also[edit]

References[edit]


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