Suppose I have a summation of the form $\sum\limits_{n=0}^\infty n(n-1) c_n x^{n-2}$. As is done in my textbook, we can replace $n$ with $n + 2$ and begin the summation at $n = 2$ instead of $n = 0$, giving us $\sum\limits_{n=2}^{\infty} (n+2)(n+1) c_n x^{n-2}$. I'm having difficulty following this logic. Is there a general rule for rewriting the summation in this way? What if I wanted to take the summation that begins at, say, $n = 1$? How would I go backwards, in this sense?
Thanks in advance.