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Math Help - integral test help

  1. #1
    Junior Member
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    integral test help

    hello, i need help with one more problem:

    Use the Integral Test to show that


    Σ [(ln n)^k]/n^2 converges for all exponents k. You may use that
    n=2


    integral (u^k)(e^-u) du converges for all k
    1

    thanks!
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  2. #2
    MHF Contributor
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    So the integral test says that if the integral of the series converges, then the series converges as well. To integrate this, start with the substitution u=ln(n) -> du=\frac{1}{n}dn. So the numerator is simply u^2 and from the du term we have (1/n)dn There is still an extra n unaccounted for though in the denominator. So notice that e^u=n by our original substitution. Thus we can write that this integral is also \frac{u^k}{e^u}du and it seems you're allowed to simply state that this converges.
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