# Convergence of series $\sum_{n =1}^{\infty}\sin(\frac{\pi\cdot n}{4})\cdot \sqrt[9]{\ln(\frac{n+12}{n+9})}$

$$\sum_{n =1}^{\infty}\sin\left(\frac{\pi\cdot n}{4}\right)\cdot \sqrt[9]{\ln\left(\frac{n+12}{n+9}\right)}$$ How to find convergence of this series? I researched the absolute convergence and get $$\exp^{\frac{1}{3\cdot (n+9)}}$$ Thanks a lot!

• your series does converge – Dr. Sonnhard Graubner May 20 '17 at 14:56
• Please state what have you tried... – Alex Vong May 20 '17 at 14:56
• @AlexVong I researched the absolute convergence and get $\exp^{\frac{1}{3\cdot (n+9)}}$ – user448072 May 20 '17 at 15:06
• @Dr.SonnhardGraubner Can you explain why does it converge? – user448072 May 20 '17 at 15:09
• What do you mean by "I researched ..." and got that? Makes no sense. Surely you have some thoughts on this. Come on, what does the sequence $\sin(\pi n/4)$ look like? – zhw. May 20 '17 at 15:38

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 $0.$