Find the value of : $\lim_{n\to\infty}\sqrt[n]{\frac{|\sin1|}1+\cdots+\frac{|\sin n|}{n}\ }$ I just read this question, about a limit very similar to that I am asking. I was confused because I was misreading the product dots in that question as plus signs. The provided, excellent answers are easy to follow, and in fact they allow me to realize about my mistake. Now I am curious about the limit
$$\lim_{n\to\infty}\sqrt[n]{\frac{|\sin1|}1+\cdots+\frac{|\sin n|}{n}\ }\,.$$
I did not try anything, sorry, my only intuition is that the inner sum probably diverges, so its $n$-th root has indeterminate behavior

Upper bound (since $|\sin k|  \leq 1$:
$$
L \leq \lim_{n \to \infty} \bigg( \sum_{k=1}^{n} \frac{1}{k} \bigg)^\frac{1}{n} \sim\lim_{n \to \infty} (\log n)^\frac{1}{n} = \lim_{n \to \infty}e^{\frac{\log \log n}{n}}=1
$$
Lower bound:
$$
L \geq \lim_{n \to \infty}\bigg(\frac{1}{n}\bigg)^\frac{1}{n}=\lim_{n \to \infty} e^{-\frac{\log n}{n}}=1
$$
Now use the squeeze lemma
