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So my doubt is for this comment of this video:

Is ANYONE going to notice the fact that the kid's solution to the problem given at 9:35 is completely invalid? I mean, $\sin(2n)/(1+\cos^4(n))$ fails to be either positive or monotonically decreasing, meaning it fails ALL the conditions of the Integral Test, making his solution completely invalid. To make matters worse, the corresponding function is straight-up periodic, which means that a quick limit to infinity and invocation of the $N$th Term Test would've sufficed to show it FAILS to converge.

If you could come up with a more explicit example would appreciate it, including both a valid example of the question as whether if the child example perhaps could came to a feasible point of what he wanted to do, thanks.

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  • $\begingroup$ What exactly is your question? It seems to me that the comment is right, and the kid is wrong. If we let $n \to \infty$, the terms don't converge to $0$, so how can the series converge? $\endgroup$
    – TMM
    May 5, 2012 at 13:31
  • $\begingroup$ wolframalpha.com/input/?i=sin%282*n%29%2F%281%2Bcos%5E4%28n%29%29 $\endgroup$ May 5, 2012 at 13:40
  • $\begingroup$ @dato's link fixed: WolframAlpha $\endgroup$
    – TMM
    May 5, 2012 at 13:50

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