Calculate the following limit
$$\displaystyle\lim_{n\to\infty}\sqrt[n]{(\sin1)^2+(\sin2)^2+...+(\sin n)^2}$$
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2The limit makes not much sense because it does not depend on $x$, resp. it's just the term right of $\lim$. – emacs drives me nuts Sep 08 '22 at 14:05
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sorry，the x should change into n – Tesla Sep 08 '22 at 14:17
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We know that
$$ \sum_{k=1}^{n} \sin^2(k) = \frac{n}{2} + \frac{1-\csc(1) \sin(2n+1)}{4} $$ And since $|\sin(x)| \le 1$ we can say that $$ \frac{n}{2} + C_1 <\sum_{k=1}^{n} \sin^2(k) < \frac{n}{2} + C_2 $$ where $C_{1}, C_2$ are some constants. Can you conclude from here?
Robert Lee
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