Series involving a Logarithm

Question

Consider the series \begin{align} \sum_{n=1}^{\infty} \left[ \frac{n}{a} \ln\left(1 + \frac{a}{n}\right) - 1 + \frac{a}{2n} \right]. \end{align} Is there a closed form solution to this series and what is the value when $a=1, 2$ ?

Answer 1

For small $|a|$, $\ln(1 + a/n) = \sum_{j=1}^\infty (-1)^{j-1} (a/n)^j/j$, so your series is

$$ \sum_{n=1}^\infty \sum_{j=2}^\infty (-1)^{j} \dfrac{a^{j}}{(j+1)\; n^{j}} = \sum_{j=2}^\infty \dfrac{(-a)^{j}}{j+1} \zeta(j)$$

I don't think this has a closed form.