Summation of Series

Evaluating sums.

The Wolfram System recognizes this as the power series expansion of (e^x):

Sum[x^n/n!,{n,0,Infinity}]

This sum comes out in terms of a Bessel function:

Sum[x^n/(n!^2),{n,0,Infinity}]

Here is another sum that can be done in terms of common special functions:

Sum[n! x^n/(2n)!,{n,1,Infinity}]

Generalized hypergeometric functions are not uncommon in sums:

Sum[x^n/(n!^4),{n,0,Infinity}]

There are many analogies between sums and integrals. And just as it is possible to have indefinite integrals, so indefinite sums can be set up by using symbolic variables as upper limits.

This is effectively an indefinite sum:

Sum[k,{k,0,n}]

This sum comes out in terms of incomplete gamma functions:

Sum[x^k/k!,{k,0,n}]

This sum involves polygamma functions:

Sum[1/(k+1)^4,{k,0,n}]

Taking the difference between results for successive values of n gives back the original summand:

FullSimplify[%-(%/.n->n-1)]

The Wolfram System can do essentially all sums that are found in books of tables. Just as with indefinite integrals, indefinite sums of expressions involving simple functions tend to give answers that involve more complicated functions. Definite sums, like definite integrals, often, however, come out in terms of simpler functions.

This indefinite sum gives a quite complicated result:

Sum[Binomial[2k,k]/3^(2k),{k,0,n}]

The definite form is much simpler:

Sum[Binomial[2k,k]/3^(2k),{k,0,Infinity}]

Here is a slightly more complicated definite sum:

Sum[PolyGamma[k]/k^2,{k,1,Infinity}]