$\sum_{n=1} ^\infty \frac{8}{n15^n}$ without calculator I just can't figure out which function to use to get the sum. I tried with ln, but that gives me an alternating series.

The exchange between integral and series is justified because the convergence is uniform when far from $1$. 
$$
\sum_{n=1} ^\infty \frac{8}{n15^n}
=8\,\sum_{n=1}^\infty \int_0^{1/15}t^{n-1}\,dt
=8\,\int_0^{1/15}\left(\sum_{n=1}^\infty t^{n-1}\right)\,dt\\
=8\,\int_0^{1/15}\frac{dt}{1-t}=-8\,\log(1-t)\left.\vphantom{\int}\right|_0^{1/15}\\
=-8\,\log\left(\frac{14}{15}\right)
$$
