Originally Posted by

**gdmath** Its known that:

$\displaystyle

e^x=\sum_{n=0}^{\infty} \frac{x^n}{n!} \quad \quad (1)$

Now if we integrate both parts we have that:

$\displaystyle

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

$\displaystyle

e^x=\int\frac{x^0}{0!}+\int\frac{x^1}{1!}+...\Righ tarrow$

and correct me if i am wrong but:

$\displaystyle

\int\frac{x^n}{n!}=\frac{x^{n+1}}{(n+1)!}$

So

$\displaystyle

e^x=\frac{x^1}{1!}+\frac{x^2}{2!}+...\Rightarrow$

$\displaystyle

e^x=\sum_{n=0}^{\infty} \frac{x^{n+1}}{(n+1)!}\quad \quad (2)$

And comparing 1 and 2

$\displaystyle

\sum_{n=0}^{\infty} \frac{x^{n}}{(n)!}=\sum_{n=0}^{\infty} \frac{x^{n+1}}{(n+1)!}$

**What am i missing here?**

Thank you a lot