Prove that sin (n^2) + sin (n^3) is not a convergent

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SUMMARY

The sequence defined by \(\sin(n^2) + \sin(n^3)\) is proven to be non-convergent due to the oscillatory nature of the sine function. The discussion highlights that unlike simpler sequences such as \(\sin(n)\), the combination of \(\sin(n^2)\) and \(\sin(n^3)\) does not settle towards a limit as \(n\) approaches infinity. The values of \(n\) such as 3, 31, 314, 3141, and 31415 illustrate the lack of convergence in the sequence.

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Kummer
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Prove that \sin (n^2) + \sin (n^3) is not a convergent sequence.
 
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Unpleasant!
 
how about just sin(n). that's already interesting. as opposed to sin(pi n) for example.
 
mathwonk said:
how about just sin(n).
I believe that is easier. Consider n=3,31,314,3141,31415,...
 

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