Consider the power series
with coefficients determined by the identity
Compute the coefficients and determine the sum of the series.
We know the power-series expansion for is given by
Starting with and the given identity we can compute the coefficients by equating the coefficients of like powers of ,
Then from the identity for the coefficients (and noting that the series converges absolutely for all real so we may split the sum into separate sums without worry),
This is a first order linear differential equation of the form . Furthermore, the initial condition implies that when . Therefore, the solution is
So, we have