I'm stuck on this.. Given $\sum_{n=1}^{\infty}\frac{n!}{n^n}\cdot sin(n^2)$ I have to determine if it's convergent or not.
I can see that $sin(n^2)$ is bounded and so are the partial sums of it. But $\frac{n!}{n^n}$ doesnt converge to $0$ to use Dirichlet's test
Also my intuition is that it's not convergence but I can't find another divergent sequence to use the comparison test
Any hints?
Best Answer
Hint $$\frac{n!}{n^n}=\frac{1}{n}\cdot\frac{2}{n}\cdot...\cdot\frac{n}{n}\leq \frac{1}{n}\cdot\frac{2}{n}\cdot 1 \cdot 1 \cdot... \cdot 1=\frac{2}{n^2}$$