evaluate series and express sum with harmonic numbers I've computed the series but have had trouble expressing the sum in harmonic numbers
For $M\geq1$ compute the sum of the series below at $x=\dfrac{1}{\sqrt{M}}$ and express the sum in harmonic numbers.  e.g $3H + H $ where 
 $3H=3\sum_{k=1}^{n}\dfrac{1}{k}$
$$  \sum_{n=1}^\infty \dfrac{x}{n(1+x^2n)}$$

Substitute $x=\frac{1}{\sqrt{m}}$ into the sum & do partial fractions
\begin{eqnarray*}
\sum_{n=1}^{\infty} \frac{\frac{1}{\sqrt{M}}}{n(1+\frac{n}{M})}=\sqrt{M} \sum_{n=1}^{\infty} \frac{1}{n(n+M)} = \frac{1}{\sqrt{M}}\sum_{n=1}^{\infty} \left( \frac{1}{n} -\frac{1}{n+M} \right) =\color{red}{ \frac{H_M}{\sqrt{M}}}.
\end{eqnarray*}
