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I Summation algebra

  1. Mar 21, 2016 #1
    Hi, I'm working with series solutions of differential equations and I have come across something that has troubled me other courses as well. given that
    \begin{equation}
    \sum_{n=0}^{\infty} c_{n+2}x^n+e^{-x} \sum_{n=0}^{\infty}c_{n}x^n \\
    \text{where}\\
    e^{-x}=\frac{1}{\sum_{n=0}^{\infty}\frac{x^n}{n!}}\\
    \sum_{n=0}^{\infty} c_{n+2}x^n+\frac{ \sum_{n=0}^{\infty}c_{n}x^n }{\sum_{n=0}^{\infty}\frac{x^n}{n!}}
    \end{equation}
    now my problem is I have the xn in every term and the limits are the same, but I have a fraction of sums and I want to a way to make it simpler. can the xn cancel each other in the fraction. if so
    i then have
    \begin{equation}
    \sum_{n=0}^{\infty} c_{n+2}x^n+ \sum_{n=0}^{\infty} c_n*n!
    \end{equation}
    Is this even allowed?
     
  2. jcsd
  3. Mar 21, 2016 #2

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    No, this is not correct.

    You can see this even by just looking at three terms: there is no reason why
    ##\frac{c_0+c_1x +c_2x²}{1+x+x²/2}=c_0+c_1 +c_2 2##
    would be correct in general.
     
  4. Mar 21, 2016 #3

    Mark44

    Staff: Mentor

    No. Cancelling works when the same factor appears in both numerator and denominator.
     
  5. Mar 21, 2016 #4
    Hi crazycool2:

    NO!

    Take a look at making your cancellation with respect to a similar finite sum.

    (a +bx+cx2) / (1 + x/1 + x2/2) =? (a+b+c)/(1+1+1/2)​

    I understand that this insight will not help you simplify the DE solution you have. I am not sure what kind of simplification you need, but I suggest you start by combining the two sums into one. To do this first rewrite the first sum to be from 2 to ∞.

    Hope this helps.

    Regards,
    Buzz
     
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