$$\sum_{n =1}^{\infty}\sin\left(\frac{\pi\cdot n}{4}\right)\cdot \sqrt[9]{\ln\left(\frac{n+12}{n+9}\right)}$$ How to find convergence of this series? I researched the absolute convergence and get $$\exp^{\frac{1}{3\cdot (n+9)}}$$ Thanks a lot!
2026-03-25 19:51:34.1774468294
Convergence of series $\sum_{n =1}^{\infty}\sin(\frac{\pi\cdot n}{4})\cdot \sqrt[9]{\ln(\frac{n+12}{n+9})}$
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Hints: The series does not converge absolutely. Proof idea: Sum over the indices $n=2,10,18,26,\dots.$ The series does converge conditionally. Proof idea: The partial sums of $\sum \sin(\pi n/4)$ are bounded and the terms $[\ln ((n+2)/(n+9))]^{1/9}$ decrease to $0.$