Consider the following series $$S_N = \sum_{k=1}^N \frac{\sin[k\pi/(N+1)]}{k}.$$ What is the asymptotic behavior of this series for $N \to \infty$?
If we replace the sine function with the factor $1$, we get the standard harmonic series which diverges $\sim \ln N$. However, if we replace the sine function with $(-1)^k$, we get a convergent series with the result $\ln 2$. Therefore, I think that $S_N$, defined above, should diverge, but more slowly than the logarithm. Am I correct?
It is a Riemann sum, hence $$ S_N = \frac{1}{{N + 1}}\sum\limits_{k = 1}^N {\frac{{\sin (\pi k/(N + 1))}}{{k/(N + 1)}}} \to \int_0^1 {\frac{\sin (\pi x)}{x}dx} = 1.8519370\ldots $$