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Infinite Series

  1. Mar 13, 2006 #1
    The sum of a series:
    [tex]\sum _{n=0} ^{\infty} \frac{2^{2n+1}x^{2n}}{n!}[/tex]
    is:
    a)[tex]2cos(2x)[/tex]
    b)[tex]cos(x^2)[/tex]
    c)[tex]e^{2x}[/tex]
    d)[tex]2e^{2x^2}[/tex]
    e) None of the above.


    I have absolutely no idea how I would go about solving this. I know various tests for convergence and divergence, but the only methods I have to calculate sums are the geometric sum method, and the approximation method. I'm not sure how to solve this.

    Any help is greatly appreciated, Thank You.
     
  2. jcsd
  3. Mar 13, 2006 #2

    StatusX

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    Do you know about taylor series? Specifically, what is the taylor series for ex?
     
  4. Mar 13, 2006 #3
    No I haven't learned the Taylor Series. Is that the only way of doing this question?
     
  5. Mar 13, 2006 #4
    What's the sum of the series for x = 0? How does it compare to the values of the listed functions at
    x = 0?
     
  6. Mar 14, 2006 #5

    HallsofIvy

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    It also might help you to note that the sum is never negative.
     
  7. Mar 14, 2006 #6
    Does this require knowledge of Talor Series?
     
  8. Mar 14, 2006 #7

    VietDao29

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    No, it does not.
    As Archon has pointed out in post #4, if x = 0, then what's the value of the sum? Is there any of the listed functions which returns the same value as the sum of that series at x = 0?
    Can you go from here?
     
  9. Mar 14, 2006 #8

    StatusX

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    You might be able to use the process of elimination here, but that's not a very useful approach in general. And how could you eliminate the none of the above option?

    The only other way I can think of is to find a differential equation this series solves and then find the analytic solution to it. That is, defining the series as a function y(x), find an equation relating y to y' and then solve for y(x).
     
    Last edited: Mar 14, 2006
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