Sum

Author

Eric W. Weisstein
January 11, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/Sum.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Sum.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Geometric Series

FullSimplify[Sum[x^k,{k,0,Infinity}]^p]

FullSimplify[1/(p-1)!Sum[(n+p-1)!/n! x^n,{n,0,Infinity}]]

FullSimplify[1/(p-1)!Sum[Pochhammer[n+1,p-1] x^n,{n,0,Infinity}]]

Favard Constants

f=Sum[((-1)^k/(2k+1))^(r+1),{k,0,Infinity}]

favard=Table[4/Pi f/Pi^r,{r,0,100}];

Take[favard,6]

Zeigenbein's Square Identity

s[n_]:=Sum[k,{k,n}]

s[10]

10^2-s[9]

10^2-(9^2-s[8])

10^2-(9^2-(8^2-s[7]))

{Abs[FoldList[#2^2-#1&,0,Range[10,1,-1]]],Table[(-1)^k s[k],{k,10,0,-1}]
}

Plus@@%

Factor[n^2-n(n-1)/2]