In mathematics, an alternating series is an infinite series whose terms alternate between positive and negative:
![{\displaystyle 1\;-\;{\frac {1}{2}}\;+\;{\frac {1}{4}}\;-\;{\frac {1}{8}}\;+\;{\frac {1}{16}}\;-\;{\frac {1}{32}}\;+\;\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f580104890f5243f64219e94c180077cc62d9bf3)
Any two adjacent terms in an alternating series must have opposite signs.
- Grandi's series:
![{\displaystyle 1\,-\,1\,+\,1\,-\,1\,+\,1\,-\,1\,+\,\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5bb18aa54d7d919f3a021be0a2c05a3e5af43ee)
This series diverges, though Leibniz and others have argued that the proper value of the sum is 1⁄2.
- The alternating harmonic series:
![{\displaystyle 1\;-\;{\frac {1}{2}}\;+\;{\frac {1}{3}}\;-\;{\frac {1}{4}}\;+\;{\frac {1}{5}}\,-\,{\frac {1}{6}}\,+\,\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6aa17360648e3667c1f12e0c61263214510a94d)
This series converges to ln 2 ≈ 0.69314718. The sum of just the positive terms of this series is infinite, as is the sum of just the negative terms. (Such a series is called conditionally convergent.)
- The Leibniz series for pi:
![{\displaystyle 1\;-\;{\frac {1}{3}}\;+\;{\frac {1}{5}}\;-\;{\frac {1}{7}}\;+\;{\frac {1}{9}}\,-\,\cdots \;=\;{\frac {\pi }{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60c77c85661bd885f0da6b331681a4112ba60e9e)
- Geometric series with negative common ratio:
![{\displaystyle a\,-\,ar\,+\,ar^{2}\,-\,ar^{3}\,+\,\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4db494497c60a7e846920269553f095f552a169b)
This category includes divergent series such as 1 − 2 + 4 − 8 + · · ·, and convergent series such as 1/2 − 1/4 + 1/8 − 1/16 + · · ·.
When written as a summation, alternating series are often expressed with a (−1)n in the formula, since this alternates between −1 and +1:
![{\displaystyle (-1)^{0}=1\qquad (-1)^{1}=-1\qquad (-1)^{2}=1\qquad (-1)^{3}=-1\qquad \cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/da2540daf293f45868d02653d6df72f149be17f8)
For example:
![{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2^{n}}}\;=\;1\;-\;{\frac {1}{2}}\;+\;{\frac {1}{4}}\;-\;{\frac {1}{8}}\;+\;{\frac {1}{16}}\;-\;{\frac {1}{32}}\;+\;\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc16b231cf0a6337367141531844d4a25297aa9e)
![{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}\;=\;1\;-\;{\frac {1}{3}}\;+\;{\frac {1}{5}}\;-\;{\frac {1}{7}}\;+\;{\frac {1}{9}}\;-\;{\frac {1}{11}}\;+\;\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/69d108b708b5320b2a48e1c98806e41949af7756)
When using a (−1)n, the terms with even values of n are positive, and the terms with odd values of n are negative. If the opposite signs are required, a (−1)n−1 can be used instead:
![{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}\;=\;1\;-\;{\frac {1}{2}}\;+\;{\frac {1}{3}}\;-\;{\frac {1}{4}}\;+\;{\frac {1}{5}}\;-\;{\frac {1}{6}}\;+\;\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3165ba7721f704eb24e488b01a1efb7d866506d)
The alternating series test (or Leibniz test, named after Gottfried Leibniz) provides a simple criterion for proving the convergence of an alternating series. In many cases, an alternating series converges even though the corresponding series of positive numbers would diverge—such a series is called conditionally convergent.