Finding the coefficient of $x^r$ in exponential series: $\frac{e^{nx}-1}{1-e^{-x}}$

Question

Find the coefficient of $x^r$ in the expansion of $$\frac{e^{nx}-1}{1-e^{-x}}$$ I try to simplify it but I was no susessful .so I try to merely expand it but I found it so difficult to compute $x^r$ term . so please anyone give me hint to solve it .thanks

Answer 1

Hint. One may recall that $$ \frac{u^n-1}{u-1}=1+u+\cdots+u^{n-1}, \qquad u\neq 1, $$ giving $$ \frac{e^{nx}-1}{1-e^{-x}}=e^x\cdot\frac{e^{nx}-1}{e^x-1}=e^x+e^{2x}+\cdots+e^{nx}, \qquad e^x\neq 1, $$ then one may recall that $$ e^{ax}=\sum_{k=0}^\infty\frac{a^k}{k!}x^k. $$