Results 1 to 4 of 4

Math Help - The integral test and decreasing functions...

  1. #1
    Newbie
    Joined
    Feb 2010
    From
    Canada
    Posts
    7

    The integral test and decreasing functions...

    hello,

    I am studying the integral test at the moment and I understand that in order to be able to use the integral tests on series to find whether they converge or diverge is by making sure that the function is continuous, positive and decreasing.

    I am having problem with proving that the function is decreasing...

    for example i got the following problem:




    I know that I could use something called the first derivative test which requires me to take the derivative of the function and then find the critical points but I am not sure how that can help me prove that the function decreases... Any help would be greatly appreciated.

    thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Ted
    Ted is offline
    Member
    Joined
    Feb 2010
    From
    China
    Posts
    193
    For f(x)=\frac{x}{2x^3+1}, Clearly its positive and continuous for n \geq 1.
    Now, we need to show that it is decreasing.
    What is your derivative ?
    Also, What is question exactly? Is "test the series for convergence" or "use the integral test to ..." ?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Feb 2010
    From
    Canada
    Posts
    7
    The derivative equals



    and the question simply asks to find whether the series converges or diverges.

    thanks.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Ted
    Ted is offline
    Member
    Joined
    Feb 2010
    From
    China
    Posts
    193
    Quote Originally Posted by dmitrip View Post
    The derivative equals



    and the question simply asks to find whether the series converges or diverges.

    thanks.
    Well, Clearly -4x^3+1=1-4x^3 < 0 \,\ \forall x \geq 1
    Hence, the numerator is negative, and clearly the denominator is positive, It follows that f'<0 for x \geq 1.
    And this proves the function is decreasing.
    Now, focus on the convergence/divergence of \int_1^{\infty} \frac{x}{2x^3+1}dx.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Increasing / Decreasing Test
    Posted in the Calculus Forum
    Replies: 9
    Last Post: May 31st 2011, 02:21 AM
  2. Test functions as a taylor series with integral remainder
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: May 13th 2011, 08:53 AM
  3. increasing and decreasing functions
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 15th 2010, 02:11 AM
  4. Integral (continious decreasing functions)
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 6th 2010, 05:31 PM
  5. Decreasing Functions
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 19th 2009, 10:09 AM

Search Tags


/mathhelpforum @mathhelpforum