Alternating series

In mathematics, an alternating series is an infinite series of the form

with a_n > 0 for all n. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.

The geometric series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ sums to 1/3.

The alternating harmonic series has a finite sum but the harmonic series does not.

The Mercator series provides an analytic expression of the natural logarithm:

The functions sine and cosine used in trigonometry can be defined as alternating series in calculus even though they are introduced in elementary algebra as the ratio of sides of a right triangle. In fact,

When the alternating factor (–1)ⁿ is removed from these series one obtains the hyperbolic functions sinh and cosh used in calculus.

For integer or positive index α the Bessel function of the first kind may be defined with the alternating series

If s is a complex number, the Dirichlet eta function is formed as an alternating series

that is used in analytic number theory.

The theorem known as "Leibniz Test" or the alternating series test tells us that an alternating series will converge if the terms a_n converge to 0 monotonically.

Proof: Suppose the sequence ${\displaystyle a_{n}}$ converges to zero and is monotone decreasing. If ${\displaystyle m}$ is odd and ${\displaystyle m<n}$, we obtain the estimate ${\displaystyle S_{n}-S_{m}\leq a_{m}}$ via the following calculation:

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