Undergrad Simplifying Summation Algebra with Differential Equations

[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
\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?

 

[Samy_A]

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?

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.

 

[Mark44]

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?

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

 

[Buzz Bloom]

Is this even allowed?

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