Convergence of $\sum \frac{\sin(n^2)}{n}$ Does the numerical series $\sum \frac{\sin(n^2)}{n}$ converges ?
For the moment I have tried a discrete integration by parts but it involves the asymptotic behaviour of $\sum \sin(n^2)$ which seems complex.
Trying a comparison with an integral does not seem very useful too.
 A: By Weyl's inequality we have
$$ \sum_{n=1}^{N}\sin(n^2)\ll N^{\frac{7}{12}} \log^2(N) $$
hence the given series is convergent by summation by parts.
A: Attempts to calculate the limit
Now that Jack D'Aurizio has proved the convergence of the sum 
$$s_{2,1} = \sum_{n=1}^\infty \frac{\sin(n^2)}{n}$$
we ask for the value of the infinite sum.
To this end we compare numerical approaches using Mathematica 8.0
1) Accumulate[]
Here a table of values for all $n$ to be considered is created first, then the patial sums are calculated using Accumulate[].
A strange kinky structure in the (unfinished) approach to a limit appears, hereby shedding doubt on the method and the result.
$$ \text{Accumulate[]}(s_{2,1})\simeq 0.165\tag{1}$$
 
2) NSum[] with no specified method
$$\text{NSum[]}(s_{2,1}) = 0,1785031 \tag{2}$$
Warnings:   

NIntegrate::deodiv: DoubleExponentialOscillatory returns a finite
  integral estimate, but the integral might be divergent.
NSum::emcon: Euler-Maclaurin sum failed to converge to requested error
  tolerance.

3) NSum[] with method AlternatingSigns
The graph now looks promising, and the limit calculated as the average over the last $10^5$ values is
$$\text{NSum[altsigns]}(s_{2,1}) = \text{0.10446481678}\tag{3}$$

4) Check of method AlternatingSigns
Let us check if the method of 3) is reliable with a case in which the exact result of the limit is known
$$s_{1,1} = \sum_{n=1}^\infty \frac{\sin(n)}{n} = \frac{\pi - 1}{2}\simeq 1.0708$$
Unfortunately, as shown in the graph the numeric result differs significantly from the exact value.
Hence we are somewhat lost in the middle of (unappropriate) numerical methods.

5) van der Corput
As suggested by Jack D'Aurizio the method of van der Corput (https://en.wikipedia.org/wiki/Van_der_Corput%27s_method) which was specifically designed to generate estimates for exponential sums should be tried.
