Editing Alternating series (section)

==Examples==
The geometric series [[1/2 − 1/4 + 1/8 − 1/16 + ⋯|{{sfrac|1|2}} − {{sfrac|1|4}} + {{sfrac|1|8}} − {{sfrac|1|16}} + ⋯]] sums to {{sfrac|1|3}}.

The [[harmonic series (mathematics)#Alternating harmonic series|alternating harmonic series]] has a finite sum but the [[harmonic series (mathematics)|harmonic series]] does not. The series
<math display="block">1-\frac{1}{3}+\frac{1}{5}-\ldots=\sum_{n=0}^\infty\frac{(-1)^n}{2n+1}</math>
[[Leibniz formula for π|converges to]] <math>\frac{\pi}{4}</math>, but is not absolutely convergent.

The [[Mercator series]] provides an analytic [[power series]] expression of the [[natural logarithm]], given by
<math display="block"> \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n  = \ln (1+x),\;\;\;|x|\le1,
x\ne-1.</math>

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]].
<math display="block">\sin x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}</math> and
<math display="block">\cos x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!} .</math>
When the alternating factor {{math|(–1)<sup>''n''</sup>}} is removed from these series one obtains the [[hyperbolic function]]s 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
<math display="block"> J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \, \Gamma(m+\alpha+1)} {\left(\frac{x}{2}\right)}^{2m+\alpha} </math> where {{math|Γ(''z'')}} is the [[gamma function]].

If {{mvar|s}} is a [[complex number]], the [[Dirichlet eta function]] is formed as an alternating series
<math display="block">\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</math>
that is used in [[analytic number theory]].