I want to see if this series converges or not:
$$
\sum_{n=1}^\infty n^{-1/2}\sin(n)\sin(n^2).
$$
I tried comparison tests but nothing. I saw that integral criteria works but I don't know how to show that.
Thank you
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1$\begingroup$ Convergence would follow if sums of the form $\sum_{n=1}^M \sin(n)\sin(n^2)$ had an upper bound independent of $M$. This is certainly known for sums $\sum_{n=1}^M \sin(n)$. I'm sure it's also true for these sums, but I don't have an immediate proof. $\endgroup$– Todd TrimbleCommented Nov 16, 2020 at 1:34
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1$\begingroup$ I'm pretty sure using basic trig and quadratic identities allows us to reduce @ToddTrimble 's method to whether $\sum_{n = 1}^M \sin((n + a)^2)$ and $\sum_{n = 1}^M \cos((n + a)^2)$ are bounded. That said, I suspect that while the question may not be immediately obvious, it seems unlikely to be asked with research-level understanding. $\endgroup$– user44191Commented Nov 16, 2020 at 4:15
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3$\begingroup$ Sounds like one of those "it's either trivial, or depends on the irrationality measure of pi being 2" questions :D $\endgroup$– Dror SpeiserCommented Nov 16, 2020 at 7:33
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9$\begingroup$ Well, indeed the sum telescopizes nicely, but the joy is immediately killed by the suspicion of homeworkness $\endgroup$– Pietro MajerCommented Nov 16, 2020 at 7:52
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2$\begingroup$ Thank you everyone for help. I'm training for some exams and I found it. I lost like 3 hours trying to solve it. Now I got it, thank you $\endgroup$– ursuv2Commented Nov 16, 2020 at 22:12
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1 Answer
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As indicated by Todd Trimble in comments, we can use the Dirichlet test; here, since $$\sin(n)\sin(n^2)=\frac12\big( \cos n(n-1) - \cos n(n+1) \big)$$ we have a telescopic sum $$\sum_{n=1}^M \sin(n)\sin(n^2)=\frac12-\frac12 \cos M(M+1)=\sin^2\Big(\frac{M(M+1)}2\Big),$$ that does not exceed $1$ in absolute value.
answered Nov 16, 2020 at 7:48