Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

At what point do the sums of the reciprocals converge?

  1. Aug 17, 2011 #1
    It is well known that the Harmonic Series diverges (1/1+1/2+1/3+1/4+...), but that the series [1/1+1/4+1/9+1/16+...] converges. In the first series, the denominators are the integers, whereas in the second example, the denominators are the integers to the power of 2. My question is, at what power do the sums of the reciprocals "switch" from divergence to convergence?
     
  2. jcsd
  3. Aug 17, 2011 #2

    phyzguy

    User Avatar
    Science Advisor

    The integral test for convergence tells us that the infinite sum:
    [tex]\sum_1^\infty f(n)[/tex]
    and the integral:
    [tex]\int_1^\infty f(n)[/tex]
    either both converge or both diverge. Since:
    [tex]\int_1^\infty \frac{1}{x^{1+\delta}} = \frac{1}{\delta}[/tex]
    This tells us that the infinite series:
    [tex]\sum_1^\infty \frac{1}{x^{1+\delta}}[/tex]
    will converge as long as delta is greater than zero, no matter how small it is.
     
  4. Aug 17, 2011 #3
    Wow, simple as that. Thanks!!
     
  5. Aug 17, 2011 #4
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...