Proving that the sum of (u^x)/x from 0 to infinity = e^u

[CraigH]

Can someone please show me how the formula

\sum^{∞}_{0}\frac{u^{x}}{x!} = e^{u}

Is derived?

Or link me to an explanation.

Thanks!

http://www.wolframalpha.com/input/?i=sum+of+%28%28a%5Ex%29%2F%28%28x%29%21%29%29+from+x%3D0+to+x+%3D+inf
(Just to show you what I'm talking about)

 

[tiny-tim]

Hi CraigH!

Isn't that the definition of e^u ?

(and you can check it by working out e^ue^v, or by differentiating it )

 

[dextercioby]

Do you know about the MacLaurin series expansion ? What is it for e^x ?

 

[CraigH]

Ah yes, I've just looked up the MacLaurin series for e^x in my data book, its give me the answer i was hoping for. I needed to know as part of a proof for a different equation.

So in my exam just saying:

"From the MacLaurin series:
The sum of (u^x)/x! from 0 to infinity = e^u"

Should be enough then.

Thank you for answering.