Question

Dose `sum ((n^2+3)/(2+n^2))^(n^3)` with `n = 0 ->` infinity converge ?

Answer 1

No.

Explanation:

Use the n-th term test:

For any integer `n`, `(n^2+3)/(2+n^2)` will always be greater than 1.

Therefore, `((n^2+3)/(2+n^2))^(n^3)` will always be greater than 1 for any positive integer `n`.

This means that we can be sure that:

`lim_(n->oo)((n^2+3)/(2+n^2))^(n^3) >= 1`

Which means that `lim_(n->oo)((n^2+3)/(2+n^2))^(n^3) != 0` for sure.

The series fails the n-th term test, and therefore diverges.