Show that for positive integer \(n\), \(x^n - y^n = (x-y)\displaystyle\sum_{r=1}^{n} x^{n-r} y^{r-1}\).
Let \(\mathrm{F}\) be defined by
\[ \mathrm{F}(x) = \frac{1}{x^n(x-k)} \quad \text{for } x \neq 0,\, k \]
where \(n\) is a positive integer and \(k \neq 0\).
Given that
\[ \mathrm{F}(x) = \frac{A}{x-k} + \frac{\mathrm{f}(x)}{x^n}, \]
where \(A\) is a constant and \(\mathrm{f}(x)\) is a polynomial, show that
\[ \mathrm{f}(x) = \frac{1}{x-k}\left(1 - \left(\frac{x}{k}\right)^n\right). \]
Deduce that
\[ \mathrm{F}(x) = \frac{1}{k^n(x-k)} - \frac{1}{k}\sum_{r=1}^{n} \frac{1}{k^{n-r}x^r}. \]
By differentiating \(x^n \mathrm{F}(x)\), prove that
\[ \frac{1}{x^n(x-k)^2} = \frac{1}{k^n(x-k)^2} - \frac{n}{xk^n(x-k)} + \sum_{r=1}^{n} \frac{n-r}{k^{n+1-r}x^{r+1}}. \]
Hence evaluate the limit of
\[ \int_2^N \frac{1}{x^3(x-1)^2} \; \mathrm{d}x \]
as \(N \to \infty\), justifying your answer.