Demonstrate that the derivative of the power series of e^x, it's its own power seriesby xvtsx Tags: demonstrate, derivative, power, series 

#1
May810, 10:24 AM

P: 15

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? 



#2
May810, 11:06 AM

HW Helper
P: 1,495

Carry out the differentiation explicitly.




#3
May810, 11:25 AM

P: 15

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




#4
May810, 11:26 AM

HW Helper
P: 1,495

Demonstrate that the derivative of the power series of e^x, it's its own power series
The n factorial is just a constant. The differentiation is with respect to x.




#5
May810, 11:34 AM

P: 15

Okay I just got a weird answer, which I think its wrong. [tex]\frac{\mathrm{d} }{\mathrm{d} x}=\frac{(n!)}{nx^{n1}}[/tex] could you give some steps cause for me its weird to differentiate explicitly with n and factorial.




#6
May810, 11:36 AM

HW Helper
P: 1,495

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!?




#7
May810, 11:55 AM

P: 15

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




#8
May810, 11:57 AM

HW Helper
P: 1,495

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.




#9
May810, 12:09 PM

P: 15

Oh sorry. The only thing I can say is this dx/dx= n(x^n1)(1)/n!




#10
May810, 12:15 PM

HW Helper
P: 1,495

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.




#11
May810, 12:32 PM

P: 15

Honestly, I dont see it. what should I consider f(x) and g(x) ? because I only see n(x^n1)(1)/n! as f(x).Sorry if I cause you trouble..




#12
May810, 12:39 PM

HW Helper
P: 1,495

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.



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