Results 1 to 2 of 2

Thread: The Integral Test for infinite series

  1. #1
    Senior Member
    Jan 2009

    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

    ∫ f(x) dx converges.

    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 (, it looks like there is no assumption of continuity in the integral 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)]
    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.
    But for improper integrals,

    ∫ f(x) dx
    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

    ∫ f(x) dx diverges ?

    Thank you for clarifying!

    [also under discussion in math links forum]
    Last edited by kingwinner; Mar 14th 2010 at 02:14 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Aug 2007
    Calm. Think it through.

    f(k) can't be continuous.

    f(x) must be integrable - Riemann Integrable.

    Does that say anything about continuity?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 10
    Last Post: Mar 22nd 2011, 06:27 PM
  2. Replies: 1
    Last Post: Aug 2nd 2010, 05:04 PM
  3. infinite series--integral?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Mar 22nd 2010, 07:11 AM
  4. The definite integral as an infinite series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Jun 7th 2009, 03:13 AM
  5. [SOLVED] Infinite Series test
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Mar 23rd 2009, 05:39 PM

Search Tags

/mathhelpforum @mathhelpforum