Results 1 to 2 of 2

Math Help - Integral Test/Comparison Test

  1. #1
    Member
    Joined
    Nov 2008
    Posts
    75

    Integral Test/Comparison Test

    infinity over Sigma n=1 n^2/n^3 + 1 Use the integral test to see if the infinite series converges.





    infinity over Sigma n=2 n^2+1/n^3.5-2 Use either the Comparison Test or the Limit Comparison Test to see if the infinite series converges.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    11
    Quote Originally Posted by Link88 View Post

    infinity over Sigma n=1 n^2/n^3 + 1 Use the integral test to see if the infinite series converges.
    Put f(x)=\frac{{{x}^{3}}}{{{x}^{2}}+1}, then f is a positive, continuous and decreasing function for x\ge2, hence, the integral test applies and the series will converge or diverge if \int_{2}^{\infty }{\frac{{{x}^{3}}}{{{x}^{2}}+1}\,dx} does.

    Since \frac{{{x}^{3}}}{{{x}^{2}}+1}\ge \frac{{{x}^{3}}}{{{x}^{2}}+{{x}^{2}}}=\frac{1}{2}x  , then the integral diverges, so does the series. (Actually, one can bound immediately the general term of the series rather than applying the integral test.)

    -----

    As for your second question, compare the series with \sum\limits_{n=2}^{\infty }{\frac{1}{{{n}^{1,5}}}} which is a convergent p-series with p=1,5>1.

    So, put {{a}_{n}}=\frac{{{n}^{2}}+1}{{{n}^{3,5}}-2} and {{b}_{n}}=\frac{1}{{{n}^{1,5}}}, and observe that \underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n}}}{{{b}_{n}}}=1, so your series converges.
    Last edited by Krizalid; January 27th 2009 at 11:12 AM. Reason: Adding the solution for question #2.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Comparison test and Limit Comparison test for series
    Posted in the Calculus Forum
    Replies: 5
    Last Post: November 25th 2010, 12:54 AM
  2. Comparison or Limit Comparison Test Problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 12th 2010, 07:46 AM
  3. Integral Test and comparison test help!
    Posted in the Calculus Forum
    Replies: 6
    Last Post: June 3rd 2009, 09:46 PM
  4. Limit comparison/comparison test series
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 25th 2009, 08:27 PM
  5. Comparison & Limit Comparison test for series
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 25th 2009, 04:00 PM

Search Tags


/mathhelpforum @mathhelpforum