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Could you help me answer the question, if there exists $\alpha <1$ such that series $n^{-\alpha} \sum _{k=1}^n \sin (k^2)$ is bounded?

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Maybe it may help to know that the integral sinx² is convergent? (integral test) That might help for your question? –  imranfat Jun 6 '13 at 20:46
    
Integral test does not help at all, since the terms are not positive. –  i707107 Jun 6 '13 at 21:34
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Try Dirichlet's test. –  Mhenni Benghorbal Jun 6 '13 at 21:42
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@MhenniBenghorbal: Can you put it as an answer? I want to see how Dirichlet's test would apply to this problem. –  i707107 Jun 6 '13 at 21:53
    
math.stackexchange.com/questions/2270/… Related –  i707107 Jun 6 '13 at 22:25
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