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I need to determine if the two series are converge/converge absolutely/diverge: $1.\sum^{\infty}_{n=2}\frac{\sin{n}}{n\sqrt{n}}+\frac{\cos{n}}{n\ln(n)} $.

$2.\sum^{\infty}_{n=1}n^{\frac{1}{4}}(\sqrt{n^{3}+1}-\sqrt{n^{3}-1})\cos n$

I prove that the first series converge using Abel's test, but I can't determine if it is converge absolutely. For the second series I have no idea what test I should use or which function to compare. Thanks for the help!

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For the second series, rationalise the numerator of the contents of parentheses: $$\Bigl\lvert n^{\frac14}(\sqrt{n^3+1}-\sqrt{n^3-1})\cos n\Bigr\rvert\le n^{\frac14}(\sqrt{n^3+1}-\sqrt{n^3-1})\begin{aligned}[t]&=\frac{2 n^{\frac14}}{\sqrt{n^3+1}+\sqrt{n^3-1}}\\&\sim_\infty\frac{2 n^{\frac14}}{2n^{\frac32}}=\frac1{n^\frac54},\end{aligned} $$ which converges, hence the second series converges absolutely.

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