I must look pathetic, but I'm beginning with infinite series.

I invented to myself a very easy infinite series that I must determine its convergence. (I'm sure it converges even if I'm not able to prove it!).

The series is : .

I tried the ratio test, writing down and , dividing by I finally got . Taking the limit when tends to I got . As , the series diverges. Where is my mistake? Also, I would like to know if I can know to what number it converges.