Question Show that the series $\sum _{n=1}^{\infty }\dfrac {nz^{n-1}} {z^{n}-\left( 1+\dfrac {1} {n }\right) ^{n}}$ converges absolutely if $\left| z\right| < 1$.
Answer For, when $\left| z\right| < 1$,\left| z^{n}-\left( 1+\dfrac {1} {n}\right) ^{n}\right| \geq \left( 1+\dfrac {1} {n}\right) ^{n}-\left| z^{n}\right| \geq 1+1+\dfrac {n-1} {2n}+\ldots -1>1 , so the moduli of the terms of the series are less than the corresponding terms of the series $\sum _{n=1}^{\infty }n\left| z^{n-1}\right| $; but this latter series is absolutely convergent, and so the given series converges absolutely.
I am having a trouble following the reason how the first 2 inequalities in the solution hold. I am sure this must be a trivial question, but i am dumb. Any help would be much appreciated.