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!

Homework Help: Compare a series sum to it's equivelent integral

  1. Sep 24, 2007 #1
    1. The problem statement, all variables and given/known data

    excuse my formatting.


    sum(from n=2 -> infinity) of 5/(n^1.1)


    integral (from 1 -> infinity) of 5/(x^1.1)

    2. Relevant equations

    3. The attempt at a solution

    if it was sum and integrate from 2 it would be easy... the sum is a rienmann sum using right endpoints and delta-x of 1, and it's error will always be larger than it's integral on a decreasing curve.

    however because the function is decreasing, and the integral starts at 1, not 2, i think i have to calculate the size of the error of the rienmann compared to it's integral from 2 -> infinity, and compare that to the integral from 1 ->2.

    maybe i'm just having a brain coniption, but i cant seem to sum either the rienmann or the integral from 2 -> infinity... anyone have any suggestions?


  2. jcsd
  3. Sep 24, 2007 #2
    It's easier to see what to do with a simpler function. Let's use 1/x.

    Draw a graph, using bars of width 1 to represent each value, so you end up with a bar graph of 1/x. The sum of the area under the bars is the sum of 1/x. Now notice that you can have two curves, one joining up the top left corners of the all bars, and one joining up the top right corners. These curves are 1/x curves, but one of them is shifted along by 1 unit. The area under these curves can be found by integrating 1/x, between appropriate limits. Can you see how to apply this method to your problem?
  4. Sep 24, 2007 #3
    it's always something simple isnt it..

    area [tex]\frac{5}{x^{1.1}} * (x - (x-1)) < \int_{x-1}^x\frac{5}{x^{1.1}}dx[/tex]

    for all values of positive x, so the sum must be smaller than the integral

    thanks for the tip

    Last edited: Sep 24, 2007
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook