How to calculate this infinite sum?

Whenever I type in the format

symsum(1/nchoosek(2k,k))),1,inf)

It gives me the error: "Unexpected MatLab Expression". What do I do to fix this problem?

[Youssef Khmou]

the function nchoosek is working with numeric data.

Answer 1

[Roger Stafford]

Thanks to Leonhard Euler we can express atan(z) as the infinite series:

   atan(z) = symsum(2^(2*n)*(n!)^2/(2*n+1)!*z^(2*n+1)/(1+z^2)^(n+1),n,0,inf)

See:

   http://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Infinite_series

From this is it possible to derive the power series:

   4*atan(sqrt(x/(4-x)))/sqrt(x)/(4-x)^(3/2)+1/(4-x) = ...

   symsum((n!)^2/(2*n)!*x^(n-1),n,1,inf)

Hence by setting x = 1 in this series we get the answer to your question:

    symsum((n!)^2/(2*n)!,n,1,inf) = 2*pi/9/sqrt(3)+1/3

Note: (n!)^2/(2*n)! is equal to 1/nchoosek(2*n,n).

Apparently your symbolic toolbox was unable to go through this line of reasoning. It is possible to verify this result numerically provided an appropriate method is used for computing the sum of the infinite series. Directly computing (2*n)! and n! will fail for large values of n.