41: Limit algebra does not help, since there is the oscillating sine in the denominator, which does not have a limit. Does intuition help us any here? Note that in the numerator, the sine is not added to some other term, that is, we cannot hope to claim that it is negligible compared to something else and thus ignore it. In fact, since the sine is multiplied there, it allows us to try another thing: pull it out of the fraction. So the given sequence is of the form "oscillating sine times a certain expression". The outcome is then given by the behaviour of this expression, the fraction that is left after pulling the sine out. Check it out.

By the way, note that we do not have sin(n) here, but a sine of a more complicated expression. This means that the argument of sine only picks some natural numbers, not all of them, so it might conceivably happen that these points fall to the places where the sine waves are small. To put it another way, {sin(n² + 1)} is a subsequence of {sin(n)}, so while the latter is oscillating and divergent, it might happen that the subsequence actually converges somewhere. This is really hard to investigate, so we hope that the answer we obtain when investigating the fraction as outlined above will allow us to ignore this complication.

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