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Taylor series

The Taylor series for polynomes is

 

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Explanation

Take the polynome P_n(x) of degree n. The development in powers of x − a has the form

Multiple differentiation of P_n(x) gives

For x = a you get

Calculation of a₀, a₁, a₂, ... , a_n and substitution in P_n shows

 

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Example 1

The exponential function has for every value of x the series development

 

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History The English mathematician Brook Taylor (1685 - 1731) is best known for the Taylor series.

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