Need help with the following:

Evaluate $\displaystyle \sum_{n=0}^{\infty }\sin \left( {\frac {1}{2^n}} \right) \cos \left( {\frac {3}{2^n}} \right) $

I recognise that the terms inside the sin and cos functions are the terms of a geometric progression with first term 1 and 3 respectively and common ratio of 0.5. But when it comes to evaluating the series with the trigonometric functions applied i'm quite lost. Is there some kind of general principle to adhere to to tackle problems of this nature?

Determine if $\displaystyle \sum _{n=0}^{\infty } \left( {\frac {3n}{4\,n+1}} \right) ^{2\,n}$ is convergent or divergent.

I'm abit confused as to which test to use for convergence/divergence for this particular problem. I'm thinking either limit or limit comparison, but I still can't seem to work it out. Would appreciate any help you guys might be able to offer.

Thanks in advance.