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 Web Resource. https://mathworld.wolfram.com/AlternatingSeries.html
Subject classifications
![sum_(n=1)^(infty)(-1)^(n+1)[e-(1+1/n)^n]](/images/equations/AlternatingSeries/Inline17.svg)
![sum_(n=1)^(infty)[(1+1/(2n))^(2n)-(1+1/(2n-1))^(2n-1)]](/images/equations/AlternatingSeries/Inline20.svg)
