Convergence of this series: $$ \sum_{n=1}^\infty \sin\Big(\frac{2x}{x+(n+5)^6}\Big)$$
x is from R
I tried different criteria and didn't get to an answer with none of them.
2026-05-06 06:54:38.1778050478
Convergence of this series: $ \sum_{n=1}^\infty \sin\Big(\frac{2x}{x+(n+5)^6}\Big)$
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For any fixed $x,$ the general term is (eventually) very close to $\sin \left((n+5)^{-6}\right) \sim \frac1{(n+5)^6}.$ And since the series $\sum_{n} \frac1{(n+5)^6} $ converges by your favorite test, you are cooking with gas.