# Taylor Polynomials

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f(x) =

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e   i   LN2   LN10   pi   SQRT2   SQRT1_2

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$$x =$$
$$f(x) =$$
$$T_n(x) =$$
Error $$=$$

## Taylor Polynomials

The Taylor series of a function $$f(x)$$ centered at $$x=a$$ is defined by $\sum_{i=0}^\infty \frac{f^{(i)}(a)}{i!}(x-a)^i = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{2!}(x-a)^3 + \cdots$ The Maclaurin series of the function $$f(x)$$ is the Taylor series centered at $$a=0$$: $\sum_{i=0}^\infty \frac{f^{(i)}(a)}{i!}(x-a)^i = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{2!}(x-a)^3 + \cdots$ The Taylor polynomial of degree $$n$$ is a partial sum of this series: $T_n(x) = \sum_{i=0}^n \frac{f^{(i)}(a)}{i!}(x-a)^i = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{2!}(x-a)^3 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$ This polynomial can be used as an approximation of the function on an interval centered around $$x=a$$. Note that if $$f(x)$$ is a polynomial of degree $$n$$, the Taylor series (and Taylor polynomial) of degree $$n$$ or higher will be exactly equal to the function.