smekkinnZuG

2022-11-25

Is
$\sum _{n=2}^{\mathrm{\infty }}\mathrm{log}\left(1+\frac{\left(-1{\right)}^{n}}{\sqrt{n}}\right)$
convergent?

Teagan Gamble

Expert

$\mathrm{log}\left(1+\frac{\left(-1{\right)}^{n}}{\sqrt{n}}\right)=\frac{\left(-1{\right)}^{n}}{\sqrt{n}}-\frac{1}{2n}+O\left(\frac{1}{n\sqrt{n}}\right)$
and since the series $\sum _{n}\frac{\left(-1{\right)}^{n}}{\sqrt{n}}$ and $\sum _{n}\frac{1}{n\sqrt{n}}$ are convergent and the series $\sum _{n}\frac{1}{n}$ is divergent then the given series is divergent.

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