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Alternating Series


A series of the form

 sum_(k=1)^infty(-1)^(k+1)a_k
(1)

or

 sum_(k=1)^infty(-1)^ka_k,
(2)

where a_k>0.

A series with positive terms can be converted to an alternating series using

 sum_(r=1)^inftyv_r=sum_(r=1)^infty(-1)^(r-1)w_r,
(3)

where

 w_r=v_r+2v_(2r)+4v_(4r)+8v_(8r)+....
(4)

Explicit values for alternating series include

sum_(k=0)^(infty)((-1)^k)/(k!)=1/e
(5)
sum_(k=1)^(infty)((-1)^(k-1))/k=ln2
(6)
sum_(k=1)^(infty)((-1)^(k-1))/(k^2)=1/(12)pi^2
(7)
sum_(k=1)^(infty)((-1)^(k-1))/(k^3)=3/4zeta(3),
(8)

where zeta(3) 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,

C=sum_(n=1)^(infty)(-1)^(n+1)[e-(1+1/n)^n]
(9)
=sum_(n=1)^(infty)[(1+1/(2n))^(2n)-(1+1/(2n-1))^(2n-1)]
(10)
=0.44562240319...
(11)

(OEIS A114884).


See also

Cahen's Constant, Dirichlet Eta Function, e, Natural Logarithm of 2, Series

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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

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