Weierstrass Approximation Theorem/Lemma 1

Theorem

Let $\map {p_{n, k} } t : \N^2 \times \closedint 0 1 \to \R$ be a real valued function defined by:

$\map {p_{n, k} } t := \dbinom n k t^k \paren {1 - t}^{n - k}$

where:

$n, k \in \N$

$t \in \closedint 0 1$

$\dbinom n k$ denotes the binomial coefficient.

Then:

$\ds \sum_{k \mathop = 0}^n k \map {p_{n, k} } t = n t$

$\ds \paren {x + y}^n$

$=$

$\ds \sum_{k \mathop = 0}^n \binom n k y^k x^{n - k}$

$\ds \leadsto \ \ $

$\ds 1$

$=$

$\ds \sum_{k \mathop = 0}^n \binom n k t^k \paren {1 - t}^{n - k}$

$y = t, ~x = 1 - t$

$\ds 0$

$=$

$\ds \sum_{k \mathop = 0}^n \binom n k \paren {k t^{k - 1} \paren {1 - t}^{n - k} - t^k \paren{n - k} \paren {1 - t}^{n - k - 1} }$

$\ds $

$=$

$\ds \sum_{k \mathop = 0}^n k p_{n,k} \paren {\frac 1 t + \frac 1 {1 - t} } - \frac n {1 - t}$

$\ds \leadsto \ \ $

$\ds n t$

$=$

$\ds \sum_{k \mathop = 0}^n k p_{n,k}$

$\blacksquare$