Alternating series

In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation this is expressed $${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}}$$ or $${\displaystyle \sum _{n=0}^{\infty }(-1)^{n+1}a_{n}}$$ with a_n > 0 for all n.

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 a_n converge to 0 monotonically, but this condition is not necessary for convergence.

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 series $${\displaystyle 1-{\frac {1}{3}}+{\frac {1}{5}}-\ldots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}}$$ converges to ${\displaystyle {\frac {\pi }{4}}}$, but is not absolutely convergent.

The Mercator series provides an analytic power series expression of the natural logarithm, given by $${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}x^{n}=\ln(1+x),\;\;\;|x|\leq 1,x\neq -1.}$$

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. $${\displaystyle \sin x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{2n+1}}{(2n+1)!}}}$$ and $${\displaystyle \cos x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{2n}}{(2n)!}}.}$$ When the alternating factor (–1)ⁿ 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 of the first kind may be defined with the alternating series $${\displaystyle J_{\alpha }(x)=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{m!\,\Gamma (m+\alpha +1)}}{\left({\frac {x}{2}}\right)}^{2m+\alpha }}$$ where Γ(z) is the gamma function.

If s is a complex number, the Dirichlet eta function is formed as an alternating series $${\displaystyle \eta (s)=\sum _{n=1}^{\infty }{(-1)^{n-1} \over n^{s}}={\frac {1}{1^{s}}}-{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}-{\frac {1}{4^{s}}}+\cdots }$$ that is used in analytic number theory.