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FoxTrot Series


The "Foxtrot series" is a mathematical sum that appeared in the June 2, 1996 comic strip FoxTrot by Bill Amend (Amend 1998, p. 19; Mitchell 2006/2007). It arose from a convergence testing problem in a calculus book by Anton, but was inadvertently converted into a summation problem on an alleged final exam by the strip's author:

 F=sum_(n=1)^infty((-1)^(n+1)n^2)/(n^3+1)=....
(1)

The sum can be done using partial fraction decomposition to obtain

F=-1/3sum_(n=1)^(infty)((-1)^n)/(1+n)+1/3sum_(n=1)^(infty)((-1)^n(1-2n))/(1-n+n^2)
(2)
=1/3(1-ln2)+(zeta^2)/3sum_(n=1)^(infty)(-1)^n(1-2n)×[1/((1+zeta)(zeta-n))+1/((1+zeta)(zeta^2+n))]
(3)
=1/3[1-ln2+pisech(1/2sqrt(3)pi)]
(4)
=0.239560747...
(5)

(OEIS A127198), where zeta=(-1)^(1/3)=(1+isqrt(3))/2 and the last sums have been done in terms of the digamma function and symbolically simplified.

FoxTrot by Bill Amend, June 2, 1996 stop. Reproduced with permission of the author.

See also

Digamma Function, Series

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References

Amend, B. Camp FoxTrot. Kansas City, MO: Andrews McMeel, p. 19, 1998.Mitchell, C. W. Jr. In "Media Clips" (Ed. M. Cibes and J. Greenwood). Math. Teacher 100, 339, Dec. 2006/Jan. 2007. Sloane, N. J. A. Sequence A127198 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

FoxTrot Series

Cite this as:

Weisstein, Eric W. "FoxTrot Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FoxTrotSeries.html

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