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Math Help - The Integral Test for infinite series

  1. #1
    Senior Member
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    The Integral Test for infinite series

    Integral test: if f is continuous(?), decreasing, and POSITIVE on [n,∞), then

    ∑ f(k) converges if and only if
    k=n

    ∫ f(x) dx converges.
    n

    To apply the integral test, does f have to be continuous? In my first year calculus text book, the integral test is as stated above; f is assumed to be continuous. But in my other textbook, there is no such assumption. The same for wikipedia (http://en.wikipedia.org/wiki/Integra...nt_of_the_test), it looks like there is no assumption of continuity in the integral test...how come?? Is it wrong?? Or maybe there is a typo?

    What is the correct statement of the integral test? Do we need to assume continuity?


    Also, I was thinking of applying integral test to determine the convergence/divergence of

    ∑ 1 / [n log n log(log n)]
    n=2
    Let f(x)=1 / [x log x log(log x)]
    Then f(2)<0, but the integral test requires f to be positive so I think I should start the integral at 3 and look at

    ∫ f(x) dx.
    3
    But for improper integrals,

    ∫ f(x) dx
    k
    Does the value of "k" here have any effect on the convergence/divergence of the improper integral?

    Is it ever possible that, for example,

    ∫ f(x) dx converges
    8

    ∫ f(x) dx diverges ?
    5

    Thank you for clarifying!


    [also under discussion in math links forum]
    Last edited by kingwinner; March 14th 2010 at 02:14 PM.
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  2. #2
    MHF Contributor
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    Calm. Think it through.

    f(k) can't be continuous.

    f(x) must be integrable - Riemann Integrable.

    Does that say anything about continuity?
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