Taylor series

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For a function, LaTeX: f, having all derivatives, the series is:

LaTeX: 
\sum_{k=0}^{\infty} \frac{f^{(k)} (x + h)}{k!}\, h^k,

where LaTeX: \textstyle f^{(k)} is the k-th derivative of LaTeX: f. Truncating the series at the n-th term, the error is given by:

LaTeX: 
\varepsilon_n (h) = \left \vert f(x) - \sum_{k=0}^{n} \frac{f^{(k)} (x + h)}{k!}\, h^k \right \vert.

This is a Taylor expansion, and for the Taylor series to equal the functional value, it is necessary that the error term approaches zero for each n:

LaTeX: 
\lim_{h \to 0} \varepsilon_n (h) = 0.

In that case, there must exist LaTeX: y in the line segment LaTeX: \textstyle [x,x+h] such that

LaTeX: 
\varepsilon_n (h) = \frac{f^{(n+1)} (y)}{(n + 1)!} h^{n+1} .


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