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Sigma with n=1 to n= positive infinity

(x^3)* (e^(-x^4))

Does it converge or diverge and use the integral test to show how. I got that it diverges, but this isn't correct. Can someone please explain? Thanks.

  • Calculus-Series -

    I assume the expression is
    (n^3)* (e^(-n^4)) for n=1 to +&inf;
    You can use the ratio test to show that it converges, namely, show that
    an+1/an < 1 ∀ k≥1.

    The integral test will result in a definite integral from 1 to +&inf; which is not an elementary function, but evaluates numerically to about 0.09. However, this still shows that the series is convergent.

  • Calculus-Series -

    Integral from 1 to infinity of
    (x^3)* (e^(-x^4)) dx =

    1/4 exp(-1)

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