Alternating series test

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In mathematical analysis, the alternating series test is the 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, Leibniz's rule, or the Leibniz criterion.

Formulation

A series of the form

$\sum _{n=0}^{\infty }(-1)^{n}a_{n}=a_{0}-a_{1}+a_{2}-a_{3}+\cdots \!$

where either all a_n are positive or all a_n are negative, is called an alternating series.

The alternating series test then says: if $|a_{n}|$ decreases monotonically and $\lim _{n\to \infty }a_{n}=0$ 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:

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

Proof

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.)

Proof of convergence

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 a_n decrease monotonically with n.

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

$a_{1}-a_{2}=S_{2}\leq S_{2m}\leq S_{2m+1}\leq S_{1}=a_{1}.$

Now, note that a₁ − a₂ is a lower bound of the monotonically decreasing sequence S_2m+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

$\lim _{m\to \infty }(S_{2m+1}-S_{2m})=\lim _{m\to \infty }a_{2m+1}=0.$

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

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.

Counterexample

All of the conditions in the test, namely convergence to zero and monotonicity, should be met in order for the conclusion to be true. For example, take the series

${\frac {1}{{\sqrt {2}}-1}}-{\frac {1}{{\sqrt {2}}+1}}+{\frac {1}{{\sqrt {3}}-1}}-{\frac {1}{{\sqrt {3}}+1}}+\cdots$

The signs are alternating and the terms tend to zero. However, monotonicity is not present and we cannot apply the test. Actually the series is divergent. Indeed, for the partial sum $S_{2n}$ we have $S_{2n}={\frac {2}{1}}+{\frac {2}{2}}+{\frac {2}{3}}+\cdots +{\frac {2}{n-1}}$ which is twice the partial sum of the harmonic series, which is divergent. Hence the original series is divergent.

See also

• Alternating series
• Dirichlet's test

References

• Konrad Knopp (1956) Infinite Sequences and Series, § 3.4, Dover Publications ISBN 0-486-60153-6
• Konrad Knopp (1990) Theory and Application of Infinite Series, § 15, Dover Publications ISBN 0-486-66165-2
• E. T. Whittaker & G. N. Watson (1963) A Course in Modern Analysis, 4th edition, §2.3, Cambridge University Press ISBN 0-521-58807-3

External links

• Weisstein, Eric W. "Leibniz Criterion". MathWorld.