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 Exercise [05.05] 
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Joined: 19 Mar 2008, 14:09
Posts: 36
Post Exercise [05.05]
We define:

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}

Therefore:

e^ae^b = \left(\sum_{n=0}^{\infty}a_n\right) \left(\sum_{n=0}^{\infty}b_n\right) = \left(\sum_{n=0}^{\infty}c_n\right)

where, c_n = a_0b_n + a_1b_{n-1} + ... + a_{n-1}b_1 + a_nb_0 which is a property of the combination of power series. Also A_n = \frac{a^n}{n!} and B_n = \frac{b^n}{n!}.

Thus:

c_0 = A_0B_0 = 1
c_1 = A_0B_1 + A_1B_0 = a + b
c_2 = A_0B_2 + A_1B_1 + A_2B_0 = \frac{b^2}{2!} + ab + \frac{a^2}{2!}=\frac{1}{2!}(a+b)^2
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c_n = \frac{b^n}{n!} + a \frac{b^{n-1}}{(n-1)!} + .... + \frac{a^{n-1}}{(n-1)!} b + \frac{a^n}{n!} = \frac{(a+b)^n}{n!}

and so we obtain:

e^ae^b = \sum_{n=0}^{\infty} c_n =  \sum_{n=0}^{\infty} \frac{(a+b)^n}{n!} = e^{(a+b)}


09 Apr 2008, 16:34
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