Editing Harmonic series (mathematics) (section)

==Related series==
{{see also|List of sums of reciprocals}}

===Alternating harmonic series===
{{See also|Riemann series theorem#Changing the sum}}
[[Image:Alternating Harmonic Series.PNG|right|thumb|240px|The first fourteen partial sums of the alternating harmonic series (black line segments) shown converging to the natural logarithm of 2 (red line).]]
The series
<math display=block>\sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots</math>
is known as the '''alternating harmonic series'''. It is [[conditional convergence|conditionally convergent]] by the [[alternating series test]], but not [[absolute convergence|absolutely convergent]]. Its sum is the [[natural logarithm of 2]].{{r|freniche}}

More precisely, the asymptotic expansion of the series begins as
<math display="block">\frac{1}{1} - \frac{1}{2} +\cdots + \frac{1}{2n-1} - \frac{1}{2n} = H_{2n} - H_n = \ln 2 - \frac{1}{4n} + O(n^{-2}).</math>
This results from the equality <math display=inline>H_n=2\sum_{k=1}^n \frac 1{2k}</math> and the [[Euler–Maclaurin formula]].

Using alternating signs with only odd unit fractions produces a related series, the [[Leibniz formula for π|Leibniz formula for {{pi}}]]{{r|soddy}}
<math display=block>\sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}.</math>

===Riemann zeta function===
{{main|Riemann zeta function}}
The [[Riemann zeta function]] is defined for real <math>x>1</math> by the convergent series
<math display=block>\zeta(x)=\sum_{n=1}^{\infty}\frac{1}{n^x}=\frac1{1^x}+\frac1{2^x}+\frac1{3^x}+\cdots,</math>
which for <math>x=1</math> would be the harmonic series. It can be extended by [[analytic continuation]] to a [[holomorphic function]] on all [[complex number]]s {{nowrap|except <math>x=1</math>,}} where the extended function has a [[simple pole]]. Other important values of the zeta function include {{nowrap|<math>\zeta(2)=\pi^2/6</math>,}} the solution to the [[Basel problem]], [[Apéry's constant]] {{nowrap|<math>\zeta(3)</math>,}} proved by [[Roger Apéry]] to be an [[irrational number]], and the "critical line" of complex numbers with {{nowrap|real part <math>\tfrac12</math>,}} conjectured by the [[Riemann hypothesis]] to be the only values other than negative integers where the function can be zero.{{r|bombieri}}

===Random harmonic series===

The random harmonic series is
<math display=block>\sum_{n=1}^{\infty}\frac{s_{n}}{n},</math>
where the values <math>s_n</math> are [[independent and identically distributed random variables]] that take the two values <math>+1</math> and <math>-1</math> with equal {{nowrap|probability <math>\tfrac12</math>.}} It converges [[Almost surely|with probability&nbsp;1]], as can be seen by using the [[Kolmogorov's three-series theorem|Kolmogorov three-series theorem]] or of the closely related [[Kolmogorov's inequality|Kolmogorov maximal inequality]]. The sum of the series is a [[random variable]] whose [[probability density function]] is {{nowrap|close to <math>\tfrac14</math>}} for values between {{nowrap|<math>-1</math> and <math>1</math>,}} and decreases to near-zero for values greater {{nowrap|than <math>3</math>}} or less {{nowrap|than <math>-3</math>.}} Intermediate between these ranges, at the {{nowrap|values <math>\pm 2</math>,}} the probability density is <math>\tfrac18-\varepsilon</math> for a nonzero but very small value {{nowrap|<math>\varepsilon< 10^{-42}</math>.{{r|schmuland|bemosa}}}}

===Depleted harmonic series===
{{main|Kempner series}}
The depleted harmonic series where all of the terms in which the digit 9 appears anywhere in the denominator are removed can be shown to converge to the value {{gaps|22.92067|66192|64150|34816|...}}.{{r|baillie}} In fact, when all the terms containing any particular string of digits (in any [[number base|base]]) are removed, the series converges.{{r|schmelzer}}