Derivation of \(S(x, m, N)\)

(a) Derive the formula for \(S(x, m, N)=\sum_{n=m}^{N} n x^{n}\) in Formula 2.16. Hint: Begin by evaluating \(S(x, m, N)-x S(x, m, N)\).

(b) Alternatively, derive the formula by noting that

\[S(x, m, N)=x \frac{d}{d x}\left(\sum_{n} x^{n}ight)\]

and use the Geometric Series Formula.

(c) Use the above formula and the fact that \(n x^{n} ightarrow 0\) as \(n ightarrow \infty\) when \(|x|<1\) to show

\[\sum_{n \geq 1} n x^{n}=\frac{x}{(1-x)^{2}} \text { when }|x|<1\]