Why do different trig functions sum differently?

Question

Why does the $\sum_{n=1}^{\infty} \sin (\frac 1 {n^2})$ converge but the $\sum_{n=1}^{\infty} \cos (\frac 1 {n^2})$ diverge?

Answer 1

$\sin t\approx t$ for small values of the argument. In our case, $t=\dfrac1{n^2}$, which approaches $0$ as n tends towards infinity. Therefore, $\displaystyle\sum_{n=1}^\infty\sin\frac1{n^2}\approx\sum_{n=1}^\infty\frac1{n^2}$ , which indeed converges, since $\displaystyle\sum_{n=1}^\infty\frac1{n^k}$ always converges for $k>1$. See harmonic series and the Basel problem for more details. On the other hand, $\displaystyle\lim_{n\to\infty}\cos\frac1{n^2}=\cos0=1\neq0$, meaning that the necessary condition for convergence is not met, hence the second series diverges.