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Finding the exact value of a summation.

  1. Feb 8, 2015 #1
    1. The problem statement, all variables and given/known data
    The sum we are given is Σ(from x=0->∞) [(x^2)(2^x)]/x!. We are asked to find the exact value of this sum using concepts discussed in class which include poisson random variables, and their expected values.

    3. The attempt at a solution

    So i know the solution to the infinite sum is divergent. I know this since:

    e^2 * Σ(from x=0->∞) (x^2) * Σ(from x=0->∞) (2^x)(e^-2)/x! = e^-2 * Σ(from x=0->∞) (x^2) since Σ(from x=0->∞) (2^x)(e^-2)/x! = 1 from poisson distribution. And we know Σ(from x=0->∞) (x^2) is divergent from simple infinite series. So the sum itself is ∞/or diverges.

    But I don't know if theres a better way to solve this using only properties of the poisson distributions and their expected values/variances instead of invoking results from infinite series.
     
  2. jcsd
  3. Feb 8, 2015 #2

    Dick

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    The sum is not divergent. A ratio test will tell you that. I don't know how to sum it exactly, but it is convergent.
     
  4. Feb 8, 2015 #3
    oh I think I noticed my error of breaking the sum up since its a property of finite sums only.
     
  5. Feb 8, 2015 #4

    LCKurtz

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    Hint: If that sum had an ##e^{-2}## factor, what expected value of what distribution would it be calculating?
     
  6. Feb 9, 2015 #5
    Yeah I kinda understand that what I'm uncertain about is how does Σ(from x=0->∞) (x^2)(2^x)(e^-2)/x! = 6. The x^2 confuses me ?
     
  7. Feb 9, 2015 #6

    LCKurtz

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    If you would answer my question it might help you.
     
  8. Feb 15, 2015 #7

    lurflurf

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    use homogeneous differentiation for these types of exercises

    $$\sum_{x=0}^\infty \dfrac{x^22^x}{x!}=\left. \left(r\dfrac{\mathrm{d}\phantom{r}}{\mathrm{d}r}\right)^2\sum_{x=0}^\infty \dfrac{r^x}{x!}\right|_{r=2}$$

    or use
    $$\sum_{x=0}^\infty \dfrac{r^x}{x!}
    =\sum_{x=0}^\infty x\dfrac{r^{x-1}}{x!}
    =\sum_{x=0}^\infty x(x-1)\dfrac{r^{x-2}}{x!}=e^x$$
     
    Last edited: Feb 15, 2015
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