Is $\sum_{k=1}^{\infty}\sin\frac{\log^{4}k}{1+tk^{3}}$ absolutely convergent for each $t>0?$

Asked Jul 04 '23 at 21:00

Active Jul 04 '23 at 21:05

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Is the series $\sum_{k=1}^{\infty}\sin\frac{\log^{4}k}{1+tk^{3}}$ absolutely convergent for each $t\in \left(0,\infty\right)$? Justify your answer.

What I did was the original series is less than $\sum_{k=1}^{\infty}\frac{\log^{4}k}{1+tk^{3}}$. I do not know which test I should apply for the convergence.

edited Jul 04 '23 at 21:05

Anne Bauval

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asked Jul 04 '23 at 21:00

maths and chess

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Does this answer your question? Convergence of $\sum_{n=1}^\infty \frac{\ln^k(n) }{n^a}$ (found using Approach$0$) – Anne Bauval Jul 04 '23 at 21:13

Is $\sum_{k=1}^{\infty}\sin\frac{\log^{4}k}{1+tk^{3}}$ absolutely convergent for each $t>0?$

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