Help with Convergence/Divergence

Question

So I am trying to prove whether the following problem converges or diverges?

$$\sum_{n=1}^\infty \left({n\over n+18}\right)^n$$

So I decided to use the Root test.

$$ L = \lim_{n\to \infty}\sqrt[n]{\left({n\over n+18}\right)^n} = \lim_{n\to \infty} {n\over n+18} = 1$$

But that answer is inconclusive, because according to the Root Test, if L $\lt 1$ than the function converges, and if L $\gt 1$, than the function diverges. But my answer is 1. Can someone please suggest some other methods through which I can determine whether the given problem converges or diverges? Thanks Alot

Answer 1

Use the limit test:

If $\lim_{n\to\infty}\left(\dfrac{n}{n+18}\right)^n$ is not $0$ or does not exist, $\sum \left(\dfrac{n}{n+18}\right)^n$ diverges. Else the test is inconclusive.

Indeed, $\sum \left(\dfrac{n}{n+18}\right)^n$ diverges.

Answer 2

Note that $\left(\frac{n+18}{n}\right)^n=\left(1+\frac{18}{n}\right)^n$. Since $\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n=e^x$, we conclude that $\left(\frac{n+18}{n}\right)^n$ has limit $e^{18}$. So the terms of our series have limit $e^{-18}$, and in particular do not approach $0$. It follows that the series diverges.