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Math Help - Convergence in Series

  1. #1
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    Convergence in Series

    Hello!

    I don't know how to determine whether a series involving a trig function converges or diverges.
    As an example, could someone show me how to determine if the following series converges?

    \sum^{\infty}_{n=1} \frac{\left|sin(n)\right|}{n^2}


    I appreciate some guidance. Cheers.

    P.S. we don't use the integral test in my course.
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  2. #2
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    Use comparison test. Note that: 0 \leq |\sin (n)| \leq 1
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  3. #3
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    Quote Originally Posted by Roam View Post
    Hello!

    I don't know how to determine whether a series involving a trig function converges or diverges.
    As an example, could someone show me how to determine if the following series converges?

    \sum^{\infty}_{n=1} \frac{\left|sin(n)\right|}{n^2}


    I appreciate some guidance. Cheers.

    P.S. we don't use the integral test in my course.
    Note that |sin(n)| < 1

    so,

    |sin(n)| / n^2 < 1/n^2

    so, does 1/n^2 converge? If so, then your original series converges by the Direct Comparison Test.
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  4. #4
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    Quote Originally Posted by coolguy99 View Post
    Note that |sin(n)| < 1

    so,

    |sin(n)| / n^2 < 1/n^2

    so, does 1/n^2 converge? If so, then your original series converges by the Direct Comparison Test.
    Yes 1/n^2 is a convergent p-series with p=2>1, I think I'm starting to get the idea now...

    Here's another kind of question that I don't understand:

    \sum^{\infty}_{n=1} sin^2(\frac{1}{n})

    What can we compare this to? We can make use of the fact that sin(n)<1 but I'm not sure what to do next...
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