Is the solution to this complex power series correct?

Question

Find all z for which the series

∑(1+i)$^n$z$^n$

converges.

Using the ratio test I got |z| < 1/(2$^1$$^/$$^2$)

Answer 1

Using the ratio test you can indeed deduce that it converges if $|z|<\frac1{\sqrt2}$. And also that it diverges if $|z|>\frac1{\sqrt2}$.

Now, it remains to see what happens if $|z|=\frac1{\sqrt2}$. In that case,$$\left|(1+i)^nz^n\right|=1$$and therefore the series diverges then, too.