convergence of series $\sum_{n=1}^{\infty} a_n$ with $|a_n| \le \frac{n+1}{n^2}$
Let us consider series:
$$S=\sum_{n=1}^{\infty} a_n$$
$$a_n=\frac{n\cos(n)+\sin(n^2+2n-5)}{n^2}$$
The bounds for $a_n$ is shown in this MSE question to be:
$$|a_n| \le \frac{n+1}{n^2}$$
Is it possible to prove that this series is convergent?
Numerical results showed that $$\sum_{n=1}^{10^5} a_n=-0.931506$$ $$\sum_{n=1}^{10^6} a_n=-0.931501$$
Here is a plot of Christian Blatter's bounds $\pm (n+1)/n^2$ and $a_n$: !enter image description here
From the plot the series, we can see that the terms are oscillating, but not like $(-1)^n |a_n| $. There seems to be some randomness in the number of terms to be positive and number of terms to be negative.