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!

Limit of a Series

  1. Apr 25, 2015 #1
    1. The problem statement, all variables and given/known data
    I'm reading a derivation and there is a step where the writer goes from:

    ## \sum_{n=0}^\infty e^{-n\beta E_0}##

    to:

    ## \frac {1} {(1-e^{-\beta E_0})}.##

    I can't see how they did this.


    2. Relevant equations

    I think it just involves equation manipulation.

    3. The attempt at a solution
    Can someone give me a clue, so I can attempt this problem?

    Kind regards.
     
  2. jcsd
  3. Apr 25, 2015 #2

    Mark44

    Staff: Mentor

    This is a geometric series.

    Questions about infinite series are normally covered in calculus courses, so I moved this thread from the Precalc section.
     
  4. Apr 25, 2015 #3
    think ##\frac{a}{1-r}##
     
  5. Apr 25, 2015 #4
    Cheers guys. I found online that "the limit of a geometric series is fully understood and depends only on the position of the number x on the real line":

    So for my case, if ##|x|\lt1,##

    then ##\sum_{n=0}^\infty x^{n}= \frac{1} {1-x}. ##

    So,

    ##\sum_{n=0}^\infty e^{-n\beta E_0} = \sum_{n=0}^\infty (e^{-\beta E_0})^n ##

    ##\Rightarrow \frac {1} {1-e^{-\beta E_0}}##
     
  6. Apr 25, 2015 #5

    Mark44

    Staff: Mentor

    Although it looks very fancy, the last line should be ##= \frac {1} {1-e^{-\beta E_0}}##. The implication arrow (##\Rightarrow##) is used to show that one statement implies the following statement. What you have instead are equal expressions.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Limit of a Series
  1. Limit of series (Replies: 6)

  2. Series Limit (Replies: 1)

  3. Limit of a series (Replies: 5)

  4. The limit of a series (Replies: 5)

Loading...