Newton Polynomial - Taylor Polynomial

Taylor Polynomial

The limit of the Newton polynomial if all nodes coincide is a Taylor polynomial, because the divided differences become derivatives.

$\lim_{(x_0,\dots,x_n)\to(z,\dots,z)} f + f\cdot(\xi-x_0) + \dots + f\cdot(\xi-x_0)\cdot\dots\cdot(\xi-x_{n-1})$

$= f(z) + f'(z)\cdot(\xi-z) + \dots + \frac{f^{(n)}(z)}{n!}\cdot(\xi-z)^n$

Read more about this topic: Newton Polynomial

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“A celibate, like the fly in the heart of an apple, dwells in a perpetual sweetness, but sits alone, and is confined and dies in singularity.”
—Jeremy Taylor (1613–1667)