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Concept
Alternating series
Summary
In mathematics, an alternating series is an infinite series of the form
or
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,
and
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:
Since is monotonically decreasing, the terms are negative. Thus, we have the final inequality: . Similarly, it can be shown that . Since converges to , 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:
That does not mean that this estimate always finds the very first element after which error is less than the modulus of the next term in the series.
Official source
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