Prove a series converges uniformly

Question

I was having trouble with this question

$$\sum_{n=0}^\infty\frac{z^n}{2^n}$$ on $D(0,1.5)$

I get you have to use the $M$-test but how would you go about doing that, any help would be so greatly appreciated

Answer 1

If $|z|<\frac32$, then $\left|\frac z2\right|<\frac34$. So $\left|\frac z2\right|^n<\left(\frac34\right)^n$, for each $n\in\mathbb{Z}^+$. Since the series $\sum_{n=0}^\infty\left(\frac34\right)^n$ converges, it follows from the $M$-test that your series converges uniformly.