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Math Help - Sum of an infinite series - comparing with a known series

  1. #1
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    Question Sum of an infinite series - comparing with a known series

    Hi,

    I'm doing a little revision and following through a sample exam paper.

    Earlier in the current question, I found the Maclaurin Series for e^x and showed that it converges.

    The last part of this question asks me to find:

    sum 3^(n-1) / n!, n = 0 to infinity
    Answer according to WolframAlpha

    In short, I can't understand how this answer is derived. I can see x = 3 for e^x series would give me something close, but not the right answer.

    It'd be great if you could offer a shove in the right direction .

    Thanks in advance,
    James
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  2. #2
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    \sum_{n=0}^{\infty} \frac{3^{n-1}}{n!}

    =\sum_{n=0}^{\infty} \frac{3^n \cdot 3^{-1}}{n!}

    =\frac{1}{3}\sum_{n=0}^{\infty} \frac{3^n}{n!}

    =\frac{1}{3} \cdot e^3 = \frac{e^3}{3} ..
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  3. #3
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    Thanks !
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