I am studying the operation going from recurrence relation to a generating function. Kind of getting the hang of it, but then the following happens:
$\sum_{n=0}^{\infty} (n+1)3^nx^n + 3x\sum_{n=0}^{\infty} (n+1)3^nx^n$
$\sum_{n=0}^{\infty} (n+1)3^nx^n + \sum_{n=0}^{\infty} (n)3^nx^n$
What is happening there?
P.S. Full link to document: https://www.math.cmu.edu/~af1p/Teaching/Combinatorics/Slides/Generating-Functions.pdf
(dia 15 )
That second sum is being re-indexed, after the extra $3x$ is multiplied inside it. Then a (value zero) first term is re-added at the front. Thus:
$$\begin{align}3x\sum_{n=0}^\infty (n+1)3^n x^n &= \sum_{n=0}^\infty (n+1)3^{n+1} x^{n+1} \\ &= \sum_{n=1}^\infty n\cdot 3^n x^n \\ &= 0+\sum_{n=1}^\infty n\cdot 3^n x^n\\ &= \sum_{n=0}^\infty n\cdot3^n x^n \end{align}$$