Weierstrass Approximation Theorem/Lemma 2

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Theorem

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

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

where:

$n, k \in \N$
$t \in \closedint 0 1$
$\dbinom n k$ denotes a binomial coefficient.


Then:

$\ds \sum_{k \mathop = 0}^n \paren {k - n t}^2 \map {p_{n, k} } t = n t \paren {1 - t}$


Proof

From the binomial theorem:

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

From Lemma 1:

\(\ds n t\) \(=\) \(\ds \sum_{k \mathop = 0}^n \binom n k k t^k \paren {1 - t}^{n - k}\)
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds \sum_{k \mathop = 0}^n \binom n k k \paren {k t^{k - 1} \paren {1 - t}^{n - k} - t^k \paren {n - k} \paren {1 - t}^{n - k - 1} }\) Definition of Derivative of Real Function with respect to $t$
\(\ds \) \(=\) \(\ds \frac 1 t \sum_{k \mathop = 0}^n k^2 \map {p_{n, k} } t - \frac n {1 - t} \sum_{k \mathop = 0}^n k \map {p_{n, k} } t + \frac 1 {1 - t} \sum_{k \mathop = 0}^n k^2 \map {p_{n, k} } t\)
\(\ds \) \(=\) \(\ds \frac 1 {t \paren {1 - t} } \sum_{k \mathop = 0}^n k^2 \map {p_{n, k} } t - \frac n {1 - t} n t\)
\(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 0}^n k^2 \map {p_{n, k} } t\) \(=\) \(\ds n t \paren {1 - t} + n^2 t^2\)

Then:

\(\ds \sum_{k \mathop = 0}^n \paren {k - n t}^2 \map {p_{n, k} } t\) \(=\) \(\ds \sum_{k \mathop = 0}^n \paren {k^2 - 2 n t k + n^2 t^2} \map {p_{n, k} } t\)
\(\ds \) \(=\) \(\ds n t \paren {1 - t} + n^2 t^2 - 2 n t n t + n^2 t^2\)
\(\ds \) \(=\) \(\ds n t \paren {1 - t}\)

$\blacksquare$


Sources

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