# I Summation algebra

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1. Mar 21, 2016

### crazycool2

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

\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!}}

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

\sum_{n=0}^{\infty} c_{n+2}x^n+ \sum_{n=0}^{\infty} c_n*n!

Is this even allowed?

2. Mar 21, 2016

### Samy_A

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.

3. Mar 21, 2016

### Staff: Mentor

No. Cancelling works when the same factor appears in both numerator and denominator.

4. Mar 21, 2016

### Buzz Bloom

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