Weierstrass Approximation Theorem/Lemma 2

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


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

where $n, k \in \N$, $t \in \closedint 0 1$, and $\displaystyle \binom n k$ stands for the binomial coefficient.

Then:
 * $\displaystyle \sum_{k \mathop = 0}^n \paren {k - nt}^2 \map {p_{n,k} } t = n t \paren {1 - t}$

Proof
From the binomial theorem:


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

From Lemma 1:

Then: