# Maclaurin and Taylor series

For a (real) function *f* under certain conditions (*Taylor's Theorem*)

*f*(*x*) = *f*(*a*) + (*x* - *a*)*f'*(*a*) + (*x* - *a*)^{2}*f ^{(2)}*(

*a*)/2! + ... + (

*x*-

*a*)

^{n}

*f*(

^{(n)}*a*)/

*n*! +

*R*

_{n}

One obtains a *Maclaurin series* when *a* = 0.*g*(*x*) = *f*(*x* + *a*)*f ^{(n)}*(

*a*) =

*g*(0), and so the Maclaurin series for

^{(n)}*g*at

*x*= 0

*f*at

*x*=

*a*.

The remainder *R*_{n} looks very much like the expected next term, with the derivative evaluated at an intermediate point:

*R*_{n} = (*x* - *a*)^{n + 1}*f ^{(n + 1)}*(

*γ*)/

*(n + 1)*!,

where γ is a point between *a* and *x*. For the derivation of this form for the remainder of the series *f* is required to have at least *n* + 1

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