A series of the form
 | (1) |
or
 | (2) |
where 
.
A series with positive terms can be converted to an alternating series using
 | (3) |
where
 | (4) |
Explicit values for alternating series include
where 
is Apéry's constant, and sums of the form (6) through (8) are special cases of the Dirichlet eta function.
The following alternating series converges, but a closed form is apparently not known,
(OEIS A114884).
See also
Cahen's Constant,
Dirichlet Eta Function,
e,
Natural Logarithm of 2,
Series Explore with Wolfram|Alpha
References
Arfken, G. "Alternating Series." §5.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 293-294, 1985.Bromwich, T. J. I'A. and MacRobert, T. M. "Alternating Series." §19 in An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, pp. 55-57, 1991.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 170, 1984.Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, p. 218, 1998.Pinsky, M. A. "Averaging an Alternating Series." Math. Mag. 51, 235-237, 1978.Shallit, J. and Davidson, J. L. "Continued Fractions for Some Alternating Series." Monatshefte Math. 111, 119-126, 1991.Sloane, N. J. A. Sequence A114884 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Alternating Series Cite this as:
Weisstein, Eric W. "Alternating Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlternatingSeries.html
Subject classifications
![sum_(n=1)^(infty)(-1)^(n+1)[e-(1+1/n)^n]](http://mathworld.wolfram.com/images/equations/AlternatingSeries/Inline17.svg)
![sum_(n=1)^(infty)[(1+1/(2n))^(2n)-(1+1/(2n-1))^(2n-1)]](http://mathworld.wolfram.com/images/equations/AlternatingSeries/Inline20.svg)
