I'm taking an introductory course on complex analysis and I've need some hints with homework.
The exercise I'm currently trying is:
Show that the series $\displaystyle\sum_{n=1}^\infty\left(\frac{z-1}{z+1}\right)^n$ is locally uniformly convergent in the semi-plane $\mathrm{Re}(z)>0$ and find the series of the sum.
The way I see it, why would this series be convergent only on $\mathrm{Re}(z)>0$? And how would I go about manually finding the series of this sum? I can't seem to find any clever way to rewrite this. Using Wolfram Alpha, I managed to find that the series converges to $\frac{z-1}{2}$