TOPICS
Alternating Series
 |
(1)
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or
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(2)
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where 
.
A series with positive terms can be converted to an alternating series using
 |
(3)
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where
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(4)
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Explicit values for alternating series include
 |  |  |
(5)
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 |  |  |
(6)
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 |  |  |
(7)
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 |  |  |
(8)
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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,
 |  | ![sum_(n=1)^(infty)(-1)^(n+1)[e-(1+1/n)^n]](https://mathworld.wolfram.com/images/equations/AlternatingSeries/Inline17.svg) |
(9)
|
 |  | ![sum_(n=1)^(infty)[(1+1/(2n))^(2n)-(1+1/(2n-1))^(2n-1)]](https://mathworld.wolfram.com/images/equations/AlternatingSeries/Inline20.svg) |
(10)
|
 |  |  |
(11)
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(OEIS A114884).
See also
Cahen's Constant, Dirichlet Eta Function, e, Natural Logarithm of 2, SeriesExplore 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 SeriesCite this as:
Weisstein, Eric W. "Alternating Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlternatingSeries.html