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Evaluate the infinite series

  1. Oct 15, 2008 #1
    1. The problem statement, all variables and given/known data

    ∑ [(e-15 15x) / x!]

    ∑ [(e-15 15x) / x!]

    2. Relevant equations

    3. The attempt at a solution
    The only way I can think of is writing out every term explicitly and adding them on a calculator.
    Is there any faster way (without having to write out every term explicitly) to calculate the above sums?

    Thanks for any help!
  2. jcsd
  3. Oct 15, 2008 #2


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    Science Advisor
    Homework Helper

    The sum of the two series is e^(-15)*e^(15), right? If you want the sums individually you do need a calculator. If you want the total, it's pretty obvious.
  4. Oct 15, 2008 #3


    Staff: Mentor

    ∑ [(e-15 15x) / x!]

    = e-15 *

    ∑ (15x) / x!

    = 1, if that's any help.
  5. Oct 16, 2008 #4


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    Staff Emeritus
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    Once you know the sum of the two series, since the first is finite, it's not all that hard to find the sum of 15x/x! for x from 0 to 15 by hand and then get the other sum by subtracting. The only place you really NEED a calculator (though I would recommend it for the tedious multiplications, divisions, and subtractions) is to evaluate e-15
  6. Oct 16, 2008 #5


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    Another approach - both of these relate to the Poisson distribution:

    \sum_{x=16}^\infty \left(\frac{e^{-15} 15^x}{x!}\right)

    is [tex] \Pr(X \ge 16) [/tex], the other sum is [tex] \Pr(X \le 15) [/tex].

    If you have access to a cumulative Poisson probability table, or to a program that will calculate these, you can save a lot of time.
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