# Thread: expansion of e^x in Taylor series

1. ## expansion of e^x in Taylor series

Am I right saying that the expansion of $\displaystyle e^x$ in Taylor series is convergent?

Thank you

2. Originally Posted by rayman
Am I right saying that the expansion of $\displaystyle e^x$ in Taylor series is convergent?

Thank you
Yes, you are.

3. thank you

4. More correctly, it is convergent for all x.

Each term in the series, about $\displaystyle x_0$, is $\displaystyle a_n= \frac{e^{x_0}}{n!}(x- x_0)^n$

Using the ratio test, $\displaystyle \left|\frac{a_{n+1}}{a_n}\right|= \frac{e^{x_0}}{(n+1)!}\left|(x- x_0)^{n+1}\right|\frac{n!}{e^{x_0}\left|(x- x_0)^n\right|}$$\displaystyle = \frac{1}{n+1}\left|x- x_0\right|$
which goes to 0< 1 for all x and all $\displaystyle x_0$.