1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integral test and series

  1. Jul 3, 2009 #1
    1. The problem statement, all variables and given/known data

    Use the integral test to determine the convergence or divergence of the series.

    [tex]\Sigma^{\infty}_{n=1}[/tex][tex]\frac{n^{k-1}}{n^{k}+c}[/tex] k is a positive integer

    2. Relevant equations



    3. The attempt at a solution

    Consider:

    [tex]\int^{/infty}_{1}[/tex][tex]\frac{x^{k-1}}{x^{k}+c} dx[/tex]

    Not sure how to integrate this expression to determine if it converges or diverges.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jul 3, 2009 #2

    zcd

    User Avatar

    This is more of a guess on my part, but since the degree of the polynomial on the denominator is only one greater than that of the numerator, the answer will be the form of natural log (i.e.[tex]\int \frac{x^0}{x^1} dx = \ln_|x| + C[/tex]). That means the integral is divergent -> series is divergent. You could also check for divergence with the limit comparison test, with a divergent series such as 1/n.
     
    Last edited: Jul 3, 2009
  4. Jul 3, 2009 #3
    What u-substitution will make this integral easy?
     
  5. Jul 3, 2009 #4
    Try letting u be the denominator and make du look like the numerator in the integrand.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook