In mathematics, an alternating series is an infinite series of the form
with an > 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
where Γ(z) is the gamma function.
If s is a complex number, the Dirichlet eta function is formed as an alternating series
that is used in analytic number theory.
Alternating series test
The theorem known as "Leibniz Test" or the alternating series test tells us that an alternating series will converge if the terms an converge to 0 monotonically.
Proof: Suppose the sequence
converges to zero and is monotone decreasing. If
is odd and
, we obtain the estimate
via the following calculation:
is monotonically decreasing, the terms
are negative. Thus, we have the final inequality:
. Similarly, it can be shown that
, our partial sums
form a Cauchy sequence (i.e. the series satisfies the Cauchy criterion) and therefore converge. The argument for
even is similar.
The estimate above does not depend on
. So, if
is approaching 0 monotonically, the estimate provides an error bound for approximating infinite sums by partial sums:
converges absolutely if the series
Theorem: Absolutely convergent series are convergent.
is absolutely convergent. Then,
is convergent and it follows that
converges as well. Since
, the series
converges by the comparison test. Therefore, the series
converges as the difference of two convergent series
A series is conditionally convergent if it converges but does not converge absolutely.
For example, the harmonic series
diverges, while the alternating version
converges by the alternating series test.
For any series, we can create a new series by rearranging the order of summation. A series is unconditionally convergent if any rearrangement creates a series with the same convergence as the original series. Absolutely convergent series are unconditionally convergent. But the Riemann series theorem states that conditionally convergent series can be rearranged to create arbitrary convergence. The general principle is that addition of infinite sums is only commutative for absolutely convergent series.
For example, one false proof that 1=0 exploits the failure of associativity for infinite sums.
As another example, by Mercator series
But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for
In practice, the numerical summation of an alternating series may be sped up using any one of a variety of series acceleration techniques. One of the oldest techniques is that of Euler summation, and there are many modern techniques that can offer even more rapid convergence.