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

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1. Aug 11, 2015

kostoglotov

1. The problem statement, all variables and given/known data

solve y" - 2xy' + y = 0

2. Relevant equations

3. 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?

2. Aug 11, 2015

DEvens

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

3. Aug 11, 2015

kostoglotov

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

4. Aug 11, 2015

Ray Vickson

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