I am reading Atkinson's "An Introduction to Numerical Analysis". I am trying to verify a limit on page 198 involving the Bernstein polynomial approximating $f(x) = x^2$.
The statement in the book is $\lim \limits_{n\to\infty} n \left( p_n(x) - f(x)\right) = x(1-x)$, where $p_n(x)$ is the Bernstein Polynomial and $f(x) = x^2$.
My work: $ p_n(x) = \sum_{k=0}^n \dbinom{n}{k} f \left( \frac{k}{n}\right) x^k (1-x)^{n-k} $
$ p_n(x) = \sum_{k=0}^n \dbinom{n}{k} \left( \frac{k}{n}\right)^2 x^k (1-x)^{n-k}$
$ n\left(p_n(x) - f(x)\right) = \left(\sum_{k=0}^n \frac{(n-1)! k }{(n-k)! (k-1)!} x^k (1-x)^{n-k}\right) - nx^2$
$ n\left(p_n(x) - f(x)\right) = x(1-x)^{n-1} + (n-1)(2)x^2 (1-x)^{n-2} + \ldots + n (x^n) (1-x) - nx^2$
Now, if I take the limit as $n$ goes to infinity, I run into the difficulty of the power of $x$ terms. Can I get some hints on how to proceed?
Thank you in advance for your help.