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Show that if $\sum_{n=1}^\infty a_n$ converges, then $\sum_{n=1}^\infty \left(\frac{1+\sin(a_n)}{2}\right)^n$ converges.
I tried to prove it using the Comparison Test and the Ratio Test but I am not able to come to a conclusive result using those two. I've tried all other convergence tests available to me but I have no luck solving this question. Any guidance would be greatly appreciated.
Hint: $|a_n| \to 0$, so for sufficiently large $n$ you'll have $|1+\sin(a_n)|/2 < 3/4$.
The idea will be if $\sum a_n < \infty$ then $a_n \to 0$, which would mean $\sin a_n \to 0$ as well, since for $x \to 0$, you have $\sin x \approx x$. So for $n > N$, each term in your series can be bounded by $$ \frac{1 + \sin(a_n)}{2} < \frac{2}{3}, $$ or any other number in $(1/2,1)$ and now your sum bounds by geometric series.
