This should be a proof of the fact that exp(x)*exp(y)=exp(x+y). Have a look at it:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\begin{align*}

\exp(x)\cdot\exp(y)&=\left(\sum_{k=0}^{\infty}\frac{x^k}{k!}\right)\cdot\left(\sum_{\ell=0}^{\infty}\frac{y^{\ell}}{\ell!}\right)\\

&=\sum_{k,\ell=0}^{\infty}\frac{x^ky^{\ell}}{k!\ell!}=\sum_{k=0}^{\infty}\left(\sum_{\ell=0}^{\infty}\frac{x^ky^{\ell}}{k!\ell!}\right)\\

&\stackrel{(\ell=n-k)}{=}\sum_{k=0}^{\infty}\left(\sum_{n=k}^{\infty}\frac{n!}{n!}\frac{x^ky^{n-k}}{k!(n-k)!}\right)\\

&=\sum_{k=0}^{\infty}\left(\sum_{n=k}^{\infty}\left(\begin{array}{*{1}{c}}n\\k\end{array}\right)\frac{x^ky^{n-k}}{n!}\right)\\

&=\sum_{k=0}^{\infty}\frac{\left(\sum_{k=0}^n\left(\begin{array}{*{1}{c}}n\\k\end{array}\right)x^ky^{n-k}\right)}{n!}\\

&=\sum_{n=0}^{\infty}\frac{(x+y)^n}{n!}=\exp(x+y)

\end{align*}

[/tex]

Now, I understand everything fairly well, except for one step: what is the operation to get from the 4th line to the 5th?

Help will be appreciated very much.

Best regards...Cliowa

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# Exponential series

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