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Math Help - Exponetial function proof

  1. #1
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    Exponetial function proof

    How to show by using Power series :
    exp(a+b)=exp(a)exp(b)

    Proof that sum ((a+b)^n)/n!=(sumsubk ((a^k)/k!))(sum subm((b^m)/m!)
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  2. #2
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    For this, you need to know the Cauchy product theorem, which says that if \sum_{m=0}^\infty a_m and \sum_{n=0}^\infty b_n are absolutely convergent series then their product is given by \Bigl(\sum_{m=0}^\infty a_m\Bigr)\Bigl(\sum_{n=0}^\infty b_n\Bigr) = \sum_{n=0}^\infty \Bigl(\sum_{k=0}^n a_kb_{n-k}\Bigr).

    If you apply that to the power series for exp(a) and exp(b) then you get \Bigl(\sum_{m=0}^\infty \frac{a^m}{m!}\Bigr)\Bigl(\sum_{n=0}^\infty \frac{b^n}{n!}\Bigr) = \sum_{n=0}^\infty \Bigl(\sum_{k=0}^n \frac{a^kb^{n-k}}{k!(n-k)!}\Bigr). We want to show that this is equal to \exp(a+b) = \sum_{n=0}^\infty \frac{(a+b)^n}{n!}. Comparing the two series, you see that this means showing that \frac{(a+b)^n}{n!} = \sum_{k=0}^n \frac{a^kb^{n-k}}{k!(n-k)!} . But if you multiply both sides by n!, that is just the binomial expansion of (a+b)^n.
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