TITLE: Using the Weierstrass M-test, show that the series converges uniformly on the given domain
QUESTION [5 upvotes]: $\sum_{k \geq 0} \frac{z^k}{z^k+1}$ on the domain $\overline{D}[0, r]$, where $0 \leq r < 1$
I'm honestly not sure how to do this.  My text mentions the Weierstrass M-test but the example they gave after stating it uses a completely different method (looks like a repeat of a previous example) and looks nothing like the M-test.

REPLY [4 votes]: From the notation and complex analysis tag I would guess we should assume $z\in \mathbb C,|z|\le r.$ For $k>0,$ we can then say $|1+z^k|\ge 1-|z^k| = 1-|z|^k \ge 1- |z| \ge 1-r.$ Thus
$$\left |\frac{z^k}{1+z^k}\right | = \frac{|z^k|}{|1+z^k|} \le \frac{r^k}{1-r}.$$
Since $\sum_{k=1}^{\infty} r^k/(1-r) < 1/(1-r)^2,$ we have uniform convergence in $|z|\le r$ by Weierstrass M.