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I Series Expansion to Function

  1. Sep 19, 2016 #1
    I ran across an infinite sum when looking over a proof, and the sum gets replaced by a function, however I'm not quite sure how.

    $$\sum_{n=1}^\infty \frac{MK^{n-1}|t-t_0|^n}{n!} = \frac{M}{K}(e^{K(t-t_0)}-1)$$

    I get most of the function, I just can't see where the ##-1## comes from. Could someone help show that?
     
  2. jcsd
  3. Sep 19, 2016 #2

    LCKurtz

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    The first term in the expansion of ##e^x## is ##1##, usually put in the sum with index ##0##. In your case the sum starts with ##n=1## so the constant term is missing on the left side. If you move the constant term from the right side to the left you will see it.
     
  4. Sep 19, 2016 #3
    Oh, not sure how I missed that. That makes sense, thanks
     
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