Summation of Power Series by Harmonic Sequence

Theorem
Consider the power series:
 * $f \left({x}\right) = \displaystyle \sum_{k \mathop \ge 0} a_k x^k$

Let $f \left({x}\right)$ converge for $x = x_0$.

Then:


 * $\displaystyle \sum_{k \mathop \ge 0} a_k {x_0}^k H_k = \int_0^1 \dfrac {f \left({x_0}\right) - f \left({x_0 y}\right)} {1 - y} \rd y$

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