1. Limited time only! Sign up for a free 30min personal 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!

Trying to find the number of terms in a series so it is accurate to 3 decimal places

  1. Feb 17, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the sum to three decimal places

    [itex]\sum^{\infty}_{n=1}[/itex][itex]\frac{1}{n^{3}}[/itex]

    2. Relevant equations



    3. The attempt at a solution

    So the following is the method that I learned how to do it.. but I think it is wrong.

    [itex]\int^{\infty}_{n}[/itex][itex]\frac{dx}{x^{3}}[/itex]

    to get

    [itex]\frac{1}{2n^{2}}[/itex]

    I then take that and do

    [itex]\frac{1}{2n^{2}}[/itex][itex]\leq[/itex]0.0005

    solving for n gets n=31.6

    so there should be 32 terms for the sum to be accurate to 3 decimal places.

    But I think I'm wrong because when I plug 32 in for n of the original function, I get something like 0.0000305. But how would anything like that small affect the third decimal place?

    Thanks!
     
  2. jcsd
  3. Feb 17, 2012 #2
    Re: Trying to find the number of terms in a series so it is accurate to 3 decimal pla

    You can use a Riemann zeta function.
     
  4. Feb 17, 2012 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Trying to find the number of terms in a series so it is accurate to 3 decimal pla

    It's because there are lots of terms after the 32nd term that have similar size. The exact value of the sum is the Riemann zeta function evaluated at 3. If you sum the first 32 terms and find the difference with zeta(3) you'll get something pretty close to your estimate.
     
  5. Feb 20, 2012 #4
    Re: Trying to find the number of terms in a series so it is accurate to 3 decimal pla

    OK thanks I didn't quite understand but now I do!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook