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I do not know exactly whether this summation $\displaystyle \sum_{n=1}^{\infty} |\sin(x_{n})|$, with $x_{n}$ approaches $0$, cannot be $\infty$ or not.

But I guess it is actually divergent but does not approach $\infty$ like $\lim \sin(n)$. I try to prove it does not go to $\infty$ by definition of limit but still not succeed.

Updated for my trouble:

If a limit diverges, there are 2 cases.

Case 1: the limit approaches infinity, like lim n.

Case 2: the limit is unidentified, like lim sin(n).

So, I just want to verify that that summation above does not always go to infinity. And, yes, I still cannot do it.

VAKK_19
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1 Answers1

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$\frac {\sin {|x_n|}} {|x_n|} \to 1$ so the series converges iff $\sum |x_n|$ converges. An example where it diverges is given by $x_n=\frac 1n$. [Note that $|\sin (x_n)|=\sin |x_n|$ at least for large $n$].

An example where it converges is given by $x_n=\frac 1{n^{2}}$.

Kavi Rama Murthy
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    I am pretty sure this answer is correct. Why is there a downvote? – Kavi Rama Murthy Oct 09 '22 at 08:59
  • I didn't downvote, because I do not care about unnecessary things like it. The answer is logically true but does not help for my trouble above. Anyway, thanks – VAKK_19 Oct 09 '22 at 09:11
  • What exactly is the trouble you are facing? You have expressed your difficulty properly. @VAKK_19 – Kavi Rama Murthy Oct 09 '22 at 09:22
  • Prove this summation cannot approach infinity. – VAKK_19 Oct 09 '22 at 09:27
  • You cannot prove something that is wrong. I have given examples to show that the series can converge or diverge depending on what $(x_n)$is . @VAKK_19 – Kavi Rama Murthy Oct 09 '22 at 09:31
  • You dont understand the problem. If a limit diverges, there are 2 cases.

    Case 1: the limit approaches infinity, like lim n.

    Case 2: the limit is unidentified, like lim sin(n).

     – VAKK_19 Oct 09 '22 at 09:31
  • And I know the limit of the sum is unidentified. But how about go to infinity? – VAKK_19 Oct 09 '22 at 09:32
  • A series of non-negative terms either increases to $+\infty$ or converges. The infinite sum always defined. It cannot oscillate. You keep saying 'the limit approaches infinity' but don't tell us limit of what approaches infinity. @VAKK_19 – Kavi Rama Murthy Oct 09 '22 at 09:35