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Power series / exponential

  1. Sep 8, 2007 #1
    1. I this from my hw solution.

    (1-t/s)^n = exp(-t/s)
    as n goes to infinity

    I don't understand. I checked the exponential power series. It should be :
    exp(x) = summation (x^n / n!)
    n=0 to infinity

    How come it could be a exponential function ?

    2. another is that why

    <t> = integral from 0 to infinity (t*P(t) dt) ?
    average t

    P(t)dt = probability that an electron has no colission till time t *
    probability that it has a collision between time t

    probablitiy has no collision is exp(-t/s)

    t+dt = exp(-t/s) *dt/s

    Thanks a lot !
  2. jcsd
  3. Sep 8, 2007 #2

    D H

    Staff: Mentor

    1. This is obviously wrong:

    [tex]\sum_{n=0}^{\infty}(1-t/s}^n \ne \exp(-t/s)[/tex]

    You don't need to expand the series. Use t/s=0.5. The series evaluates to 2, which is obviously not [itex]1/\sqrt e[/itex].

    I suspect you are missing something here.

    2. In general, the expected value of some random variable [itex]X[/itex] with respect to some probability density function [itex]p(t)[/itex] is defined as [itex]\int X p(t)dt[/itex], where the integration is performed over the domain of the PDF. Here, you want the expected value of the time until a collision for an exponentially distributed collision time. BTW, the PDF is [itex]p(t)=\exp(-t/s)/s[/tex], not [itex]\exp(-t/s)[/itex].
  4. Sep 8, 2007 #3
    oops, I should make the notation more clear.

    the ans says that

    ( 1- dt/s )^n = exp ( -t/s ) as n goes to infinity

    there's no summation here.

    I'm wondering where this comes from ?
  5. Sep 8, 2007 #4

    D H

    Staff: Mentor

    This is still not right. This is:

    [tex]\exp\left(-\;\frac t s\right) \equiv \lim_{n\to\infty} \left(1-\frac 1 n\;\frac t s\right)^n[/tex]
  6. Sep 8, 2007 #5


    User Avatar
    Staff Emeritus
    Science Advisor

    It's not a matter of "more clear"- you didn't say anything about a limit before!

    Oh, and now you have "dt/s" where before you had "t/s". Was that a typo?

    Of course, for fixed t and s, 1- t/s is just a number. If that number is larger than 1, the limit is [itex]\infty[/itex], if that number is less than 1, the limit is 0, if that number is equal to 1, the limit is 1. It certainly is not "e-t/s".

    It is true that [itex]\lim_{n\rightarrow \infty} (1+ \frac{x}{n})^n= e^{x}[/itex]
    Replacing x by -t/s, we get
    [tex]\lim_{n\rightarrow \infty} (1- \frac{x}{ns})^n= e^{-t/s}[/tex]

    Is it possible that you are missing an "n" in the denominator?
    Last edited: Sep 9, 2007
  7. Sep 8, 2007 #6
    I got it. Thanks a lot !! :)
  8. Sep 8, 2007 #7
    yes, I think so. There's an n missing in the denominator.
  9. Apr 11, 2010 #8
    hye D H...i need ur help...do check your private inbox...plssss...
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