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Math Help - alternating series 1

  1. #1
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    alternating series 1

    test the convergence or divergence for sum of (cosnpie)/n^(3/4)

    my work:
    (cosnpie)/n^(3/4) is similar to pie/(n^3/4)
    lim as n goes to infinity is (cosnpie)/n^(3/4)times(n^3/4)/pie equals infinity
    so the series diverges by limit comparison test

    is this right?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by twilightstr View Post
    test the convergence or divergence for sum of (cosnpie)/n^(3/4)

    my work:
    (cosnpie)/n^(3/4) is similar to pie/(n^3/4)
    lim as n goes to infinity is (cosnpie)/n^(3/4)times(n^3/4)/pie equals infinity
    so the series diverges by limit comparison test

    is this right?
    It actually converges:

    s_n=\frac{\cos\!\left(n\pi\right)}{n^{\frac34}}=\f  rac{\left(-1\right)^n}{n^{\frac34}} starting with n=1.

    Now, we need to apply the alternating series test.

    First, \left|s_n\right|=\left|\frac{\left(-1\right)^n}{n^{\frac34}}\right|=\frac{1}{n^{\frac3  4}}

    Since s_n>s_{n+1}\implies \frac{1}{n^{\frac34}}>\frac{1}{n^{\frac34}} is true, all we need to do is see if \lim\left|s_n\right|=0

    Thus, \lim\left|s_n\right|=\lim\frac{1}{n^{\frac34}}=0

    Therefore, the series converges.
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