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Taylor Polynomials

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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.

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This applet was last updated February 2020.