Summation of Power Series by Harmonic Sequence

Theorem
Consider the power series:
 * $\map f x = \ds \sum_{k \mathop \ge 0} a_k x^k$

Let $\map f x$ converge for $x = x_0$.

Then:


 * $\ds \sum_{k \mathop \ge 0} a_k {x_0}^k H_k = \int_0^1 \dfrac {\map f {x_0} - \map f {x_0 y} } {1 - y} \rd y$

where $H_n$ denotes the $n$th harmonic number.