calculus - Show that $sum_{n=1}^inftyarctanleft(frac{(-1)^n}{(n+1)^{0.25}}right)$ is convergent

Show that $$\sum_{n=1}^\infty\arctan\left(\frac{(-1)^n}{(n+1)^{0.25}}\right)$$ is convergent.

I'm stuck not sure what test I should use, because almost all tests require $a_n$ to be always positive which is not the case here. Do I have to test for absolute convergence instead?

Answer

Note that $\arctan\left(\frac{(-1)^n}{(n+1)^{0.25}}\right)=(-1)^n\arctan\left(\frac1{(n+1)^{0.25}}\right)$. On the other hand, $\left(\arctan\left(\frac1{(n+1)^{0.25}}\right)\right)_{n\in\mathbb N}$ is a decreasing sequence, which converges to $0$. Therefore, you can apply Dirichlet's test.