Series Soln to a Diff Eqn: can't understand one of the steps

[kostoglotov]

Homework Statement

solve y" - 2xy' + y = 0

Homework Equations

The Attempt at a Solution

in the worked example, the book gets from

here:

$$\sum\limits_{n=0}^{\infty} (n+1)(n+2)c_{n+2}x^n - \sum\limits_{n=1}^{\infty}2nc_nx^n + \sum\limits_{n=0}^{\infty}c_nx^n = 0$$

to here:

$$\sum\limits_{n=0}^{\infty} [(n+1)(n+2)c_{n+2} - (2n-1)c_n]x^n = 0$$

by way of:

$$\sum\limits_{n=1}^{\infty}2nc_nx^n = \sum\limits_{n=0}^{\infty}2nc_nx^n$$

How is this last part justified?

 

[DEvens]

What is the value of $2nc_nx^n$ when $n=0$?

 

[kostoglotov]

What is the value of $2nc_nx^n$ when $n=0$?

I considered that. It's zero. But assuming $c_n$ exists, won't $2nc_nx^n \neq 0 \ for \ n=1$?

 

[Ray Vickson]

I considered that. It's zero. But assuming $c_n$ exists, won't $2nc_nx^n \neq 0 \ for \ n=1$?

Yes, but that is irrelevant. The two sums differ by the presence/absence of the $n = 0$ term, which is zero!