Consider the series:
$$a) \sum_{n=1}^\infty \frac{a^n}{ \prod_{k=1}^n \ (1+\sin\frac{1}{2k})}$$ $$b) \sum_{n=1}^\infty \frac{a^n}{ \prod_{k=1}^n \ (1+\tan\frac{3}{2k})}$$
Showing that these two are convergent (and absolutely convergent) it's no big deal for $a\in R^*$. But I couldn't figure out how to solve it when $a=1$. My guess is that the general term has to be brought to another form. And that is my question: how to prove convergence when $a=1$.
Thanks for helping with this!
It is enough to prove the following asymptotics: $$\prod_{k=1}^{n}\left(1+\sin\frac{1}{2k}\right)\sim\prod_{k=1}^{n}\left(1+\frac{1}{2k}\right)\sim 2\sqrt{\frac{n}{\pi}} \tag{1}$$ $$\prod_{k=1}^{n}\left(1+\tan\frac{3}{2k}\right)\sim\prod_{k=1}^{n}\left(1+\frac{3}{2k}\right)\sim \frac{4}{3}\sqrt{\frac{n^3}{\pi}} \tag{2}$$ through Gautschi's inequality to get that the first series is divergent at $a=1$ while the second one is convergent, by the $p$-test. Raabe's test is another efficient alternative.
