Trying to sum a finite series I've tried to solve a financial mathematical task and I need to sum such a finite series:
$$\sum_{k=1}^{6}\frac{1.05^{3k+1}}{1.05^{3k+1}-1}$$
So I decided that maybe I'll try to find a solution to :
$$\sum_{j=1}^{n}\frac{x^{j}}{x^{j}-1}$$
My first idea was to $+1$ and $-1$ in the numerator. So I would get:
$$\sum_{j=1}^{n}1+\sum_{j=1}^{n}\frac{1}{x^{j}-1}$$
But either it is not the right approach or I just don't see how to continue it. Could somebody help me with the problem? I know I could just put the first formula to mathematica, but I couldn't do such a thing during an exam or a test, so I would like to find a paper answer.
EDIT:
I also need to find a sum of:
$$\sum_{j=1}^{n}\frac{jx^{j}}{x^{j}-1}$$
But that's even harder.

$$\sum_{n=1}^{6}\frac{1.05^{3n+1}}{1.05^{3n+1}-1}=\sum_{n=1}^{6}\frac{1}{1-(\frac{20}{21})^{3n+1}}$$
and then use
$$\frac{1}{1-x^m}=\sum_{n=0}^{\infty }x^{nm}$$
