The power series $\sum_0^\infty2^{-n}z^{2n}$ converges, if

Question

The power series $\sum_0^\infty2^{-n}z^{2n}$ converges, if

a) |$z$| $\le$ 2.

b) |$z$| $<$ 2.

c) |$z$| $\le$ $\sqrt{2}$.

d) |$z$| $<$ $\sqrt{2}$.

Please anyone give me the answer. I think option (c) is the correct.

Answer 1

$a_n = (\dfrac{z^2}{2})^n$. This is a geometric series, so it converges iff $|\dfrac{z^2}{2}| < 1$, so $|z| < \sqrt{2}$. Thus $d$. is the answer.