Converging Complex Series: Finding Limits and Sums

[Hercuflea]

Homework Statement

pardon my terrible latex skills

Find the limit of this series:

$\sum$ (n = 0 to ∞) (-1)$^{n}$($\frac{2}{3}$)$^{n}$

Homework Equations

No idea, it looks like an alternating series test, but I am supposed to actually find the sum, not just whether or not it converges.

The Attempt at a Solution

No idea

Homework Statement

$\sum$(k = 1 to ∞) $\frac{(-1)^{k}k^{3}}{(1+i)^{k}}$

Homework Equations

The Attempt at a Solution

Once again, it looks like an alternating series. I tried the root test and got (1/2)-(1/2)i, but then I realized the root test is not applicable because the series is complex. No way to compare (1/2) - (1/2)i to the real number 1 in terms of ordering. Its not geometric so I don't have a formula for finding the sum.

 

[Hercuflea]

Found the answer to the first problem,

Its a geometric series if you rewrite as Ʃ (-2/3)^n

Ratio is (-2/3) so the limit is 1/(1+2/3) = 3/5

I still have no idea how to do the second problem