Question

Show that ∞n=1 (x^n)/(n^n) converges for all x ∈ R

Answer 1

Let us show that $\sum_{n=1}^{\infty} \frac{x^n}{n^n}$ converges for all $x \in\mathbb R.$ Let us use Cauchy's root convergence test for series:

$L=\lim\limits_{n\to\infty}\sqrt[n]{|\frac{x^n}{n^n}|}=\lim\limits_{n\to\infty}\frac{|x|}{n}=|x|\cdot\lim\limits_{n\to\infty}\frac{1}{n}=|x|\cdot 0=0$

Since $L<1$ for $x\in\mathbb R,$ we conclude that $\sum_{n=1}^{\infty} \frac{x^n}{n^n}$ converges for all $x \in\mathbb R.$