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'Building Approximations for Sin(x)' printed from https://nrich.maths.org/
In general, any function whose derivatives of all orders are finite
can be written as a
power
series
$$f(x) = f(0) + x f'(0) + \frac{x^2}{2!}f''(0) + \dots +
\frac{x^n}{n!}f^{(n)}(0)+\dots$$
This expansion is an infinite series (not a polynomial).
Truncating this series at a given point provides us with a
polynomial approximation to f(x).
The question of how big the errors are in this approximation
is a difficult one to answer, and more details will be discovered
at university in Numerical Analysis courses.