This question is probably very basic, but I've been away from school for a while and the answer eludes me.

I was tempted to prove that d/dx(e^x) = (e^x) for old times sake and that was easy enough. I just expressed e^x as a power series where n goes from 0 to infinity for ((x^n)/n!).

During the derivation I started to wonder, how did they know that a power series where n goes from 0 to infinity for A(subscript n)*X^n would converge to the form of (someconstant)^x when X is 1.

Is there some theorem that proves this? IS there a more general form for X in general?

This question is pretty probably trivial for the hardcore math types but it's been bothering me, so I thought I'd ask :-) ....

This question is probably very basic, but I've been away from school for a while and the answer eludes me.

I was tempted to prove that d/dx(e^x) = (e^x) for old times sake and that was easy enough. I just expressed e^x as a power series where n goes from 0 to infinity for ((x^n)/n!).

During the derivation I started to wonder, how did they know that a power series where n goes from 0 to infinity for A(subscript n)*X^n would converge to the form of (someconstant)^x.

Is there some theorem that proves this?

This question is pretty probably trivial for the hardcore math types but it's been bothering me, so I thought I'd ask :-) ....

This question is probably very basic, but I've been away from school for a while and the answer eludes me.

I was tempted to prove that d/dx(e^x) = (e^x) for old times sake and that was easy enough. I just expressed e^x as a power series from where n goes from 0 to infinity for ((x^n)/n!).

During the derivation I started to wonder, how did they know that a power series where n goes from 0 to infinity for A(subscript n)*X^n would converge to the form of (someconstant)^x when X is 1.

Is there some theorem that proves this? IS there a more general form for X in general?

This question is pretty probably trivial for the hardcore math types but it's been bothering me, so I thought I'd ask :-) ....

This question is probably very basic, but I've been away from school for a while and the answer eludes me.

I was tempted to prove that d/dx(e^x) = (e^x) for old times sake and that was easy enough. I just expressed e^x as a power series where n goes from 0 to infinity for ((x^n)/n!).

During the derivation I started to wonder, how did they know that a power series where n goes from 0 to infinity for A(subscript n)*X^n would converge to the form of (someconstant)^x when X is 1.

Is there some theorem that proves this? IS there a more general form for X in general?

This question is pretty probably trivial for the hardcore math types but it's been bothering me, so I thought I'd ask :-) ....