# when analysis 1 becomes ugly [closed]

This is a post which comes originally from an italian forum. I report it here because it sounds extremely nice but i suspect (having done some computations myself) the solution might be extremely ugly, even though this may be a question from an analysis 1 course :)

1) Prove that $$\sum_{n=1}^\infty \frac{\sin(n^2)}{n}$$ converges,

2) Decides whether $$\sum_{n=1}^{\infty}\frac{\sin(n^2)}{\sqrt n}$$ converges or not,

3) Discuss boundedness, continuity and differentiability of the real function $$\alpha\mapsto F(\alpha)=\sum_{n=1}^\infty\frac{\sin(\alpha n^2)}{n}.$$

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–  David Speyer Oct 11 '11 at 13:11
-1 for the title. Also, if this is homework you should tell us what methods you have seen in the course. If it isn't I don't see why it has the homework tag. –  Phira Oct 11 '11 at 15:28