Does the Infinite Series Converge or Diverge? A Problem on Complex Numbers

[matpo39]

i need a little help with this problem:

determine if the infinite series converges or diverges.
summation (from n=1..infinity) {1/(n^2+i^n)}

I first applied the ration test to this series and got

(n+1)^2 + i^(n+1) / [n^2 + i^n]

i then multiplied top and bottom by (n^2 - i^n)
which gave

[{(n+1)^2 + i^(n+1) }* (n^2-i^n) ]/{n^4 - i^2n}

this is where i get stuck, i can't seem to simplify it any further.
if some one can give me some advice on it, it would be greatly appreciated.

thanks

 

[matt grime]

the ratio test is the best way to do it. but when you've got an expression like

$$\frac{(n+1)^2+c}{n^2+d}$$

all you need to do is divide every term by n^2.

 

[HallsofIvy]

Also don't forget that it is the limit of the absolute value of the ratio that counts.

 

[Muzza]

I don't think the ratio test works here. A simple comparison (of the terms' absolute values) seems to work better.