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Math Help - Power series

  1. #1
    Junior Member
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    Jul 2009
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    Power series

    Does anyone know how to find the power series of a^x ?
    and/or
    Show that e^s+t = e^se^t using algebra with power series?
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  2. #2
    Junior Member Renji Rodrigo's Avatar
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    Given two series \sum\limits^{\infty}_{k=0}a_{k} and \sum\limits^{\infty}_{k=0}b_{k}, the Cauchy product is

    \sum^{\infty}_{k=0}c_{k}
    where

    c_{n}= \sum\limits^{n}_{k=0}a_{k}b_{n-k}\;\;n \in N.

    we have
    e^x=\sum^{\infty}_{k=0}\frac{x^{k}}{k!}
    and
    e^y=\sum^{\infty}_{k=0}\frac{y^{k}}{k!}
    and by the cauchy product of e^x .e^y

    c_{n}=\sum^{n}_{k=0}\frac{x^{k}y^{n-k}}{k!(n-k)!}=\frac{(x+y)^{n}}{n!}
    so
    e^{x+y}=e^{x}.e^{y}.
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  3. #3
    Junior Member
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    Jul 2009
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    Wow thank you so much, we haven't touched on the Cauchy Product yet in class but it makes this problem much easier.
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