# Paper 4, Section II, 7D

(a) For positive integers $n, m, k$ with $k \leqslant n$, show that

$\left(\begin{array}{l} n \\ k \end{array}\right)\left(\frac{k}{n}\right)^{m}=\left(\begin{array}{l} n-1 \\ k-1 \end{array}\right) \sum_{\ell=0}^{m-1} a_{n, m, \ell}\left(\frac{k-1}{n-1}\right)^{m-1-\ell}$

giving an explicit formula for $a_{n, m, \ell}$. [You may wish to consider the expansion of $\left.\left(\frac{k-1}{n-1}+\frac{1}{n-1}\right)^{m-1} .\right]$

(b) For a function $f:[0,1] \rightarrow \mathbb{R}$ and each integer $n \geqslant 1$, the function $B_{n}(f):[0,1] \rightarrow \mathbb{R}$ is defined by

$B_{n}(f)(x)=\sum_{k=0}^{n} f\left(\frac{k}{n}\right)\left(\begin{array}{l} n \\ k \end{array}\right) x^{k}(1-x)^{n-k}$

For any integer $m \geqslant 0$ let $f_{m}(x)=x^{m}$. Show that $B_{n}\left(f_{0}\right)(x)=1$ and $B_{n}\left(f_{1}\right)(x)=x$ for all $n \geqslant 1$ and $x \in[0,1]$.

Show that for each integer $m \geqslant 0$ and each $x \in[0,1]$,

$B_{n}\left(f_{m}\right)(x) \rightarrow f_{m}(x) \text { as } n \rightarrow \infty$

Deduce that for each integer $m \geqslant 0$,

$\lim _{n \rightarrow \infty} \frac{1}{4^{n}} \sum_{k=0}^{2 n}\left(\frac{k}{n}\right)^{m}\left(\begin{array}{c} 2 n \\ k \end{array}\right)=1$