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Math Help - Convergence of series with trigonometric function :(

  1. #1
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    Convergence of series with trigonometric function :(

    hello, this is the problem:

    show that

    infinite
    Σ sin (1/n^2) is a positive, convergent series.
    n=1

    Hint: use the inequality sinx is less than or equal to x for x is greater than or equal to 0
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  2. #2
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    p-series

    Are you familiar with the convergence/divergence of the p-series \sum_{n=1}^{\infty}\frac{1}{n^p}?

    --Kevin C.
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  3. #3
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    yes, but can this be applied even though the exponent is within the trigonometric function?
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  4. #4
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    Quote Originally Posted by wyhwang7 View Post
    yes, but can this be applied even though the exponent is within the trigonometric function?
    Not exactly. First show that 1/n^2 is a converging series. Then use the hint the problem gave, that for n>0, \sin(n) \le n. Put another way, \frac{1}{\sin^2(n)} \le \frac{1}{n^2} So we have established a bounding series that converges and this proves that the other series converges as well.
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  5. #5
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    thanks so much! but if , wouldn't be the reverse? because n^2 would be the larger denominator
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  6. #6
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    I made a typo. The sine should be around the whole expression, not just the denominator. The way I wrote it was wrong, sorry.
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