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Math Help - integral test convergence divergence

  1. #1
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    integral test convergence divergence

    im confused on this one

    \sum_{n = 2}^{\infty} 6n^2 \exp(-n^3)

    i think the exponential goes to 0 does this mean that i can not use the integral test?
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  2. #2
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    Given a_n=6n^2e^{-n^3}. This is clearly positive and continuous, but is it a strictly decreasing sequence? This is the last condition so that you can apply the integral test.
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  3. #3
    Senior Member DeMath's Avatar
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    Quote Originally Posted by acosta0809 View Post
    im confused on this one

    \sum_{n = 2}^{\infty} 6n^2 \exp(-n^3)

    i think the exponential goes to 0 does this mean that i can not use the integral test?
    If you need solution

    6\int\limits_2^\infty  {{x^2}{e^{ - {x^3}}}dx}  =  - 2\int\limits_2^\infty  {{e^{ - {x^3}}}d\left( { - {x^3}} \right)}  = \left. { - 2{e^{ - {x^3}}}} \right|_2^\infty  =  - 2 \cdot \underbrace {\mathop {\lim }\limits_{x \to \infty } \frac{1}{{{e^{{x^3}}}}}}_0 + \frac{2}{{{e^{{2^3}}}}} = \frac{2}{{{e^8}}}.
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