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Homework Help: 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.
     
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