Question

The series of (3n^2+5n)/[(2^n) (n^2+1) from n=1 to n=infinity. Does the series converges or Diverges? What test can i use for that?

Answer 1

Because of the exponential, the Ratio Test is a good choice for the first attempt.

Answer 2

I get 3/2 but i ma not sure?

Answer 3

\[\sum_{n=1}^{\infty} \frac{3n^2+5n}{2^n(n^2+1)}\]
Using the Ratio Test, we have
\[\lim_{n \rightarrow \infty} \frac{3(n+1)^2+5(n+1)}{2^{n+1}[(n+1)^2+1]} \times\frac{2^n(n^2+1)}{3n^2+5n}\]
=1/2. Thus, the series converges.