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&frasl;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 &asymp; 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 (&minus;1)n in the formula, since this alternates between &minus;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 (&minus;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 (&minus;1)n&minus;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&mdash;such a series is called conditionally convergent.