Understanding the Binomial Expansion and its Relationship to e^p

[thereddevils]

How is

$$1+p+\frac{p^2}{2!}+\frac{p^3}{3!}+...=e^p$$ ?

 

[Mentallic]

I know it has to do with taylor expansions, but I've never studied this so I can't answer your question. I'd also like to see a proof for this so this is like some pointless post I'm making so I can subscribe to this thread

 

[l'Hôpital]

Depends how you define e^p. You could just define it as the power series. However, I'm assuming you are using something like...

$$e^p = \lim_{n \rightarrow \infty} (1 + \frac{p}{n})^n$$

Try using the binomial theorem on the right side, then take the limit.

 

[LCKurtz]

How is

$$1+p+\frac{p^2}{2!}+\frac{p^3}{3!}+...=e^p$$ ?

I know it has to do with taylor expansions, but I've never studied this so I can't answer your question. I'd also like to see a proof for this so this is like some pointless post I'm making so I can subscribe to this thread

You just show the remainder upon approximating it with the first n terms goes to zero as n --> infinity. See, for example,

http://en.wikipedia.org/wiki/Taylor's_theorem