Alternating series test
# Alternating series test

Part of a series of articles about :

Calculus :

Fundamental theorem

Limits

Continuity

Rolle's theorem

Mean value theorem

Inverse function theorem :

Differential

Definitions

Derivative (generalizations)

Differential

infinitesimal

of a function

total

Concepts

Differentiation notation

Second derivative

Implicit differentiation

Logarithmic differentiation

Related rates

Taylor's theorem

Rules and identities

Sum

Product

Chain

Power

Quotient

L'Hôpital's rule

Inverse

General Leibniz

Faà di Bruno's formula

Reynolds :

Integral

Lists of integrals

Integral transform

Leibniz integral rule

Definitions

Antiderivative

Integral (improper)

Riemann integral

Lebesgue integration

Contour integration

Integral of inverse functions

Integration by

Parts

Discs

Cylindrical shells

Substitution (trigonometric, tangent half-angle, Euler)

Euler's formula

Partial fractions

Changing order

Reduction formulae

Differentiating under the integral sign

Risch algorithm :

Series

Geometric (arithmetico-geometric)

Harmonic

Alternating

Power

Binomial

Taylor

Convergence tests

Summand limit (term test)

Ratio

Root

Integral

Direct comparison

Limit comparison

Alternating series

Cauchy condensation

Dirichlet

Abel :

Vector

Gradient

Divergence

Curl

Laplacian

Directional derivative

Identities

Theorems

Gradient

Green's

Stokes'

Divergence

generalized Stokes :

Multivariable

Formalisms

Matrix

Tensor

Exterior

Geometric

Definitions

Partial derivative

Multiple integral

Line integral

Surface integral

Volume integral

Jacobian

Hessian :

Advanced

Calculus on Euclidean space

Generalized functions

Limit of distributions :

Specialized

Fractional

Malliavin

Stochastic

Variations :

Miscellaneous

Precalculus

History

Glossary

List of topics

Integration Bee

Mathematical analysis

Nonstandard analysis :

vte :

In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test.

## Formal Statement

### Alternating series test

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 guarantees that an alternating series converges if the following two conditions are met:

* $|a_{n}|$ decreases monotonically, i.e., $|a_{n+1}|\leq |a_{n}|$, and
* $\lim _{n\to \infty }a_{n}=0$

### Alternating series estimation theorem

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

    $S_{k}=\sum _{n=0}^{k}(-1)^{n}a_{n}\!$

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 the alternating series test

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:

    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): S_{2(m+1)}=S_{2m}+a_{2m+1}-a_{2m+2} \geq S_{2m}

both because $a_n$ decreases 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 $a_1$ − $a_2$ 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 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 is an 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 the alternating series estimation theorem

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.

## Examples

### A typical example

The alternating harmonic series

$\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots$

meets both conditions for the alternating series test and converges.

### An example to show monotonicity is needed

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.

### The test is only sufficient, not necessary

Leibniz test's monotonicity is not a necessary condition, thus the test itself is only sufficient, but not necessary. (The second part of the test is well known necessary condition of convergence for all series.) Examples of nonmonotonic series that converge are $\sum _{n=2}^{\infty }{\dfrac {(-1)^{n}}{n+(-1)^{n}}}$ and $\sum _{n=1}^{\infty }(-1)^{n}{\dfrac {\cos ^{2}n}{n^{2}}}.$

