Please help. I am rather stuck with this problem (and our lecturer says she doesn't know how to do this either). It is part of a proof that the Gamma function is holomorphic, but was omitted from the source material the lecturer used.
Let R > 1. Show that there is some M > 0 such that $| \frac {z^{2} − z}{2} - \frac{z^{3} − z}{3j} + \frac{z^{4} − z}{4j^{2}} - ... |≤ M$ for all z ∈ $\overline{B(0, R)}$ and every integer j > R.
$\overline{B(0, R)}$ denotes the closed ball of radius R centred at 0.
Rewrite this expression (in absolute value) as: $$\sum_{k=0}^\infty \frac{z^{k+2}-z}{(k+1)j^k}$$ Throw away the first term as this won't affect boundedness of the whole series. Using or restrictions on $z$ and factoring $z^k -z$, in absolute value this is less than or equal to $$\sum_{k=1}^\infty \frac{R^{k-1}-1}{kj^{k-1}}$$ Now, there is a constant $C_k$ (dependent on $k$) such that $R^{k-1} -1 \leq C_k(R-1)^{k-1}$. I will let you find $C_k$ yourself. Using these bounds one can obtain a convergent geometric series as $R>1$ and $j>R$.