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I was wondering what is the justification for this step(changing the indexes)$\displaystyle\sum_{n=0}^{\infty}\frac{a^{n}}{n!}\sum_{m=0}^{\infty}\frac{b^{m}}{m!}=\sum_{k=0}^{\infty}\frac{1}{k!}\sum_{n=0}^{k}\frac{k!}{n!(k-n)!}a^{n}b^{k-n}$ , in Rudin's Real and complex analysis prolog to show $(\exp{a})( \exp{b})=\exp{(a+b)}$, is it the same principle that use fubini's theorem for integrals, I mean that one that says if given the domain of integration D=AxB=ExF then I can do something like $\int_{D}=\int_{A}\int_{B}=\int_{E}\int_{F}$ , I would appreciate any hint or reference to this, thanks in advance.

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    Yes you are right. It can be seen as Fubini's theorem for series.2011-09-28
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    @RagibZaman but I got a problem seeing if the "domain" of the indexes in the RHS is the same domain in the LHS of the equation. LHS is like all the plane $R^{2}$ and RHS is like the area down the identity function. Am I right?2011-09-28

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