Infinite series $\sum_{n=1}^{\infty}nx^{n+1}$ does not comply to any of my (known) tests

Question

I am attempting to find the interval of convergence for

$$\sum_{n=1}^{\infty}nx^{n+1}$$

The lower bound, x = -1, would be tested by determining if

$$\sum_{n=1}^{\infty}n(-1)^{n+1}$$

diverges.

Alternating series test results in an inconclusive answer, as do the ratio and root tests, limit test, comparison and limit comparison tests. Integral test shows that it diverges but the integral is too complex to compute for my level.

I am wondering if there is a way to show this series diverges?

Thank you

Answer 1

A necessary condition for convergence is that the terms go to $0$.