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

  1. May 8, 2010 #1
    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?
     
    Last edited: May 8, 2010
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  3. May 8, 2010 #2

    Cyosis

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

    Carry out the differentiation explicitly.
     
  4. May 8, 2010 #3
    Re: Demonstrate that the derivative of the power series of e^x, it's its own power se

    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
     
  5. May 8, 2010 #4

    Cyosis

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

    The n factorial is just a constant. The differentiation is with respect to x.
     
  6. May 8, 2010 #5
    Re: Demonstrate that the derivative of the power series of e^x, it's its own power se

    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.
     
  7. May 8, 2010 #6

    Cyosis

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

    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!?
     
  8. May 8, 2010 #7
    Re: Demonstrate that the derivative of the power series of e^x, it's its own power se

    okay. if the result its 1/n! how is that related to the power series?
     
  9. May 8, 2010 #8

    Cyosis

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

    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.
     
  10. May 8, 2010 #9
    Re: Demonstrate that the derivative of the power series of e^x, it's its own power se

    Oh sorry. The only thing I can say is this dx/dx= n(x^n-1)(1)/n!
     
  11. May 8, 2010 #10

    Cyosis

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

    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.
     
  12. May 8, 2010 #11
    Re: Demonstrate that the derivative of the power series of e^x, it's its own power se

    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..
     
  13. May 8, 2010 #12

    Cyosis

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

    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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