Math Help - calculus BC, series problem

1. calculus BC, series problem

does the series converge or diverge?
1.) the summation from 2 to infitity of [(2^k)+1]/[(3^k)+2k]

i crossed of the one and want to use the comparison test but now im stuck. since there are powers of k i want to try to use the root test but i don't know how to do it

2.) the summation from 1 to infitity of [5(sin k)^2]/(k!)
since there is a factorial i think i should use the root or ratio test but i'm not sure how to go about that either. i attempted it and got that the series converges. i may have made errors along the way though because i wasn't sure which steps to take.

any suggestions or help?? thank you in advance

2. Originally Posted by holly123
does the series converge or diverge?
1.) the summation from 2 to infitity of [(2^k)+1]/[(3^k)+2k]

i crossed of the one and want to use the comparison test but now im stuck. since there are powers of k i want to try to use the root test but i don't know how to do it

2.) the summation from 1 to infitity of [5(sin k)^2]/(k!)
since there is a factorial i think i should use the root or ratio test but i'm not sure how to go about that either. i attempted it and got that the series converges. i may have made errors along the way though because i wasn't sure which steps to take.

any suggestions or help?? thank you in advance
For the first, compare with

$\sum_{k=1}^\infty \frac{2^k}{3^k}$

For the second note that

$\frac{5 \sin^2k}{k!} \le \frac{5}{k!}$