Demonstrate that the derivative of the power series of e^x, it's its own power series

xvtsx

1. The problem statement, all variables and given/known data
I need to demonstrate that [tex]\frac{\mathrm{d} }{\mathrm{d} x}\sum_{n=0}^{\infty }\frac{x^{n}}{n!}= \sum_{n=0}^{\infty }\frac{x^{n}}{n!}[/tex]



2. Relevant equations3. The attempt at a solution

I just need a hint on how to start this problem, so how would you guys start this problem?

Cyosis

Carry out the differentiation explicitly.

xvtsx

Thanks for the quick reply, but I dont see how to take the derivative of the n factorial. could you please provide me with an example of how to do it?.Thanks

Cyosis

The n factorial is just a constant. The differentiation is with respect to x.

xvtsx

Okay I just got a weird answer, which I think its wrong. [tex]\frac{\mathrm{d} }{\mathrm{d} x}=\frac{(n!)}{nx^{n-1}}[/tex] could you give some steps cause for me its weird to differentiate explicitly with n and factorial.

Cyosis

Do you know how to differentiate x^n with n a constant? If so do you know how to differentiate constant*x^n? What if the constant equals 1/n!?

xvtsx

okay. if the result its 1/n! how is that related to the power series?

Cyosis

The result isn't 1/n!. I asked you three questions in post #6 and you avoided answering all three. If you want help you will need to cooperate.

xvtsx

Oh sorry. The only thing I can say is this dx/dx= n(x^n-1)(1)/n!

Cyosis

That is correct. Furthermore from the sum rule of differentiation you know that [itex](f(x)+g(x))'=f'(x)+g'(x)[/itex]. Therefore you can just interchange differentiation and summation. If you don't see it just write out the first few terms.

xvtsx

Honestly, I dont see it. what should I consider f(x) and g(x) ? because I only see n(x^n-1)(1)/n! as f(x).Sorry if I cause you trouble..

Cyosis

f and g are just two functions. You are dealing with a sum of more than two functions. Nevertheless the sum rule still applies in the same way and you can interchange differentiation and summation.