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Math Help - calculus BC, series problem

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    Exclamation 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
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    Quote Originally Posted by holly123 View Post
    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!}
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