Complex Series, Region of Convergence

[sgonzalez90]

Good evening, I'm an electrical engineering student questioning my answer to this series Region of Convergence problem.

Ʃ(0,inf) (n(n-1)(z+5i)^n)/n

Using the ratio test lim n-> |an+1/an|

I was able to get it down to lim n->|n(z+5i)/(n-1)| which gave |inf/inf| = 1, which means the test fails. What do I do to properly solve this? :(

 

[Office_Shredder]

you can't just write it as inf/inf and cancel them!
$$\lim_{n\to \infty} |\frac{n(z+5i)}{n-1}| = |z+5i| \lim_{n\to \infty} \frac {n}{n-1}$$

 

[sgonzalez90]

ROC
= |z+5i| < inf?

:D

 

[Office_Shredder]

Once you know the limit is |z+5i|, what does this have to be smaller than for the series to converge?