complex number

Question

the radius of convergence of the power series 

∑^∞(n +2i) zⁿ is

ⁿ⁼⁰

please sir help me please irequest you

Answer 1

I'll give this a shot.

 

The first thing you do is apply the ratio test to this series:

 

lim_n→∞ |(n+1+2i)zⁿ⁺¹ / (n+2i)z|.

 

If this limit is > 1, it diverges; if it's < 1, it converges. Simplify:

 

lim_n→∞ |(n+1+2i)/(n+2i)| |z|

 

You can put the first term in the standard complex format, but for now it should be obvious that that limit = 1, so you are left with

 

|z| < 1 for convergence.

 

That is, the radius of convergence is 1.