For $|z|\leqslant r<1$, find $M_k$ so that:
$$\frac{|z^k|}{|z^k+1|} \leqslant M_k$$
for each $k$, and $\sum M_k$ converges.
This is for the Weierstrass $M$-test. For some reason, I can't really get it straight in my head-- the numerator is clearly bounded above by $r^k$, which is promising for convergence, but the denominator is giving me trouble. The geometry tells me that $|z^k+1| \geqslant (1-r)/2$, which would suffice for my purposes, but I can't see how to get there, and I feel like there could be a more straightforward method. Could someone give me a hint?
Hint: $|1+x| \ge 1-|x|$ (Reverse triangle inequality).
Also note that $1+x$ need not be positive, or even scalar-valued. The triangle inequality holds in every normed vector space.