Monday, February 8, 2010

Math Mystery

It is amazing how easily you can get things wrong when you forget the basics. Here is a case and point from today, I was recalling the Taylor series of ex, and something didn't add up. First lets look at the intergral of ex:

$\int e^x dx = e^x$

Noting the Taylor Series of ex:

$\int e^x dx = \int \sum_{n=0}^{\infty}\frac{x^n}{n!} dx = \sum_{n=0}^{\infty} \int \frac{x^n}{n!} dx$
$=\sum_{n=0}^{\infty}\frac{x^{n+1}}{(n+1)*n!}=\sum_{n=0}^{\infty}\frac{x^{n+1}}{(n+1)!}=\sum_{n=1}^{\infty}\frac{x^{n}}{(n)!}$
$= e^x - 1$

Clearly these two methods yield a different result. Spot the error that I made, it took me a second to figure it out....