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Wikipedia / Alternating series

In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation with for all.

Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms converge to 0 monotonically, but this condition is not necessary for convergence.

Examples

The geometric series 12 − 14 + 18 − 116 + ⋯ sums to .

The alternating harmonic series has a finite sum but the harmonic series converges to, but is not absolutely convergent.

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

The functions sine and cosine used in trigonometry and introduced in elementary algebra as the ratio of sides of a right triangle can also be defined as alternating series in calculus When the alternating factor is removed from these series one obtains the hyperbolic functions sinh and cosh used in calculus and statistics.

For integer or positive index α the Bessel function where is the gamma function.

If is a complex number, the Dirichlet eta function that is used in analytic number theory.

Alternating series test

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

form a Cauchy sequence (i.e., the series satisfies the Cauchy criterion even is similar.

Approximating sums

is approaching 0 monotonically, the estimate provides an error bound also gives an alternating series where the Leibniz test applies and thus makes this simple error bound not optimal. This was improved by the Calabrese bound, discovered in 1962, that says that this property allows for a result 2 times less than with the Leibniz error bound. In fact this is also not optimal for series where this property applies 2 or more times, which is described by Johnsonbaugh error bound. If one can apply the property an infinite number of times, Euler's transform applies.

Absolute convergence

converges absolutely converges.

Theorem: Absolutely convergent series are convergent.

converges by the comparison test.

Conditional convergence

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.

Rearrangements

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. Agnew's theorem describes rearrangements that preserve convergence for all convergent series. 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

Series acceleration

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.

See also

Notes

References