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Does the series $\sum\sin^2(\frac{1}{n})$ diverge or converge?

I tried the comparison test. We have $\sin(n) \leq n$ for all natural numbers $n$, so $\sin(\frac{1}{n}) \leq \frac{1}{n}$, and $\sin^2(\frac{1}{n}) \leq \frac{1}{n^2}$ for all natural numbers $n$.

Note that the series $\sum\frac{1}{n^2}$ converges, so the series $\sum\sin^2(\frac{1}{n})$ converges by the comparison test.

What have I messed up on in my working and explanations?

Anne Bauval
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    $\sin(x)\le x$ is actually true for every non-negative real $x$ , but this has to be justified. We cannot simply replace $n$ by $\frac{1}{n}$ – Peter Dec 13 '22 at 11:53
  • @AnneBauval I have already seen that but I wanted feedback on my answer. Mine looks like the top comment they did so I think I am correct but like Peter said I am unsure on how to justify the inequality which is why I have started my own post –  Dec 13 '22 at 11:58
  • @Peter I thought you could like for the limit $\frac{sinx}{x}$ you can replace the $x$ with $\frac{1}{x}$ for example –  Dec 13 '22 at 11:59
  • @NikitaMazepin Only if the limit is for $x\to 0$ and if we let tend $x$ to $\infty$ , if we replace it by $\frac{1}{x}$ – Peter Dec 13 '22 at 12:02
  • @Peter Oh yes by a change of variables. I am allowed to state $sin(n) \leq n$ for all natural numbers $n$ if that helps, so all I need help with is justifying $sin(\frac{1}{n}) \leq \frac{1}{n}$ please –  Dec 13 '22 at 12:05
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    Your terminology is quite sloppy. It's not a series unless you write out the summation sign $\sum$. – Hans Lundmark Dec 13 '22 at 12:51
  • @HansLundmark I wrote series in place of the summation sign –  Dec 13 '22 at 13:43
  • $\sin n\le n$ for natural numbers is of no use here. Your derivation "$\sin(n) \leq n$ for all natural numbers $n$, *so* $\sin(\frac{1}{n}) \leq \frac{1}{n}$" is wrong. Use Prove $\sin(x)<x$, $\forall x>0$ or how to strictly prove $\sin x<x$ for $0<x<\frac{\pi}{2}$, applied to $x=1/n.$ – Anne Bauval Dec 13 '22 at 13:52
  • And about your previous answer to Hans Lundmark: you should not have written the word "series" "in place of" but "in addition to" the symbol $\sum.$ I correct it. – Anne Bauval Dec 13 '22 at 13:59
  • @AnneBauval okay thanks I will take a look –  Dec 13 '22 at 14:49

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