User:Jim.belk/Draft:Alternating series

In mathematics, an alternating series is an infinite series whose terms alternate between positive and negative:

$1\;-\;{\frac {1}{2}}\;+\;{\frac {1}{4}}\;-\;{\frac {1}{8}}\;+\;{\frac {1}{16}}\;-\;{\frac {1}{32}}\;+\;\cdots$

Any two adjacent terms in an alternating series must have opposite signs.

Examples

• Grandi's series:

  $1\,-\,1\,+\,1\,-\,1\,+\,1\,-\,1\,+\,\cdots$

  This series diverges, though Leibniz and others have argued that the proper value of the sum is ¹⁄₂.
• The alternating harmonic series:

  $1\;-\;{\frac {1}{2}}\;+\;{\frac {1}{3}}\;-\;{\frac {1}{4}}\;+\;{\frac {1}{5}}\,-\,{\frac {1}{6}}\,+\,\cdots$

  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:

  $1\;-\;{\frac {1}{3}}\;+\;{\frac {1}{5}}\;-\;{\frac {1}{7}}\;+\;{\frac {1}{9}}\,-\,\cdots \;=\;{\frac {\pi }{4}}$

• Geometric series with negative common ratio:

  $a\,-\,ar\,+\,ar^{2}\,-\,ar^{3}\,+\,\cdots$

  This category includes divergent series such as 1 − 2 + 4 − 8 + · · ·, and convergent series such as 1/2 − 1/4 + 1/8 − 1/16 + · · ·.

Notation

When written as a summation, alternating series are often expressed with a (−1)ⁿ in the formula, since this alternates between −1 and +1:

$(-1)^{0}=1\qquad (-1)^{1}=-1\qquad (-1)^{2}=1\qquad (-1)^{3}=-1\qquad \cdots$

For example:

$\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2^{n}}}\;=\;1\;-\;{\frac {1}{2}}\;+\;{\frac {1}{4}}\;-\;{\frac {1}{8}}\;+\;{\frac {1}{16}}\;-\;{\frac {1}{32}}\;+\;\cdots$

$\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}\;=\;1\;-\;{\frac {1}{3}}\;+\;{\frac {1}{5}}\;-\;{\frac {1}{7}}\;+\;{\frac {1}{9}}\;-\;{\frac {1}{11}}\;+\;\cdots$

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:

$\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}\;=\;1\;-\;{\frac {1}{2}}\;+\;{\frac {1}{3}}\;-\;{\frac {1}{4}}\;+\;{\frac {1}{5}}\;-\;{\frac {1}{6}}\;+\;\cdots$

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.