# convergence of a sum fails with root test

So I was trying to find the values of $p>0$ such that the sum $$\sum_{n=1}^{\infty} \left(1-\frac{p}{n}\right)^n$$

converges. I tried to use the root test but the limit of that equals $1$ so the root test is inconclusive. How else would I do this question though? I can't use the integral test or the limit comparison test. The ratio test also fails.

• From Bernoulli's Inequality, $\left(1-\frac pn\right)^n\ge 1-p$. – Mark Viola Feb 27 '18 at 1:07

It fails to converge as the $n$th terms don't tend to $0$: $$\lim_{n\to\infty}\left(1-\frac{p}{n}\right)^n = e^{-p}$$
$$\lim_{n \to \infty} \left( 1 - \frac{p}{n} \right)^n ?$$