I'm trying to find the nature of this series $$\sum_{n=0}^\infty\frac{(2+2i)}{e^{ni}n\sqrt[3]{n^2+1}}$$ but I don't know which test should I use.
2026-03-26 07:59:25.1774511965
Finding the nature of a series of complex numbers
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Assuming that the series is$$\sum_{n=0}^\infty\frac{2+2i}{e^{in}n\sqrt[3]{n^2+1}},$$it converges absolutely, since$$(\forall n\in\mathbb N):\left\lvert\frac{2+2i}{e^{in}n\sqrt[3]{n^2+1}}\right\rvert=\frac{2\sqrt2}{n\sqrt[3]{n^2+1}}\leqslant\frac{2\sqrt2}{n^{5/3}}.$$