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

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Exponential series

Loading...

Similar Threads for Exponential series | Date |
---|---|

A Closed form for series over Exponential Integral | Feb 16, 2017 |

A Convergence of an infinite series of exponentials | Nov 2, 2016 |

Exponential Power Series Expansion | Sep 16, 2011 |

F(x) of a taylor series that looks a lot like an exponential | Oct 5, 2010 |

Is the exponential series uniformly convergent? | Nov 7, 2008 |

**Physics Forums - The Fusion of Science and Community**