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The coefficient formula for a Maclaurin series is $c_n = f^{(n)}(0) / n$!. This means the nth coefficient of the power series is found by taking the nth derivative of the function, evaluating it at $x=0$, and dividing by n factorial.

Conditions: The series is expanded around $x=0$.; The function must have derivatives up to order n at 0 for the displayed formula to make sense.; The function is assumed to have a power series representation.

The coefficient formula for a Maclaurin series is $c_n = f^{(n)}(0) / n$!. This means the nth coefficient of the power series is found by taking the nth derivative of the function, evaluating it at $x=0$, and dividing by n factorial.

Conditions: The series is expanded around $x=0$.; The function must have derivatives up to order n at 0 for the displayed formula to make sense.; The function is assumed to have a power series representation.

A Maclaurin series is a special case of a Taylor series where the center of expansion is exactly 0. A Taylor series generalizes this by allowing the expansion to be centered at an arbitrary point a.

Conditions: The Taylor formula is taken with center a.; Setting $a=0$ recovers the displayed Maclaurin formula.

A Maclaurin series is a special case of a Taylor series where the center of expansion is exactly 0. A Taylor series generalizes this by allowing the expansion to be centered at an arbitrary point a.

Conditions: The Taylor formula is taken with center a.; Setting $a=0$ recovers the displayed Maclaurin formula.

When the center changes from 0 to pi, the Taylor approximations shift their region of accuracy to the neighborhood of x=pi. The quadratic approximation changes from $1-x^2/2$ to -1+(x-pi)^$2/2$.

Conditions: Function is $\cos (x)$; Center is a=pi

When the center changes from 0 to pi, the Taylor approximations shift their region of accuracy to the neighborhood of x=pi. The quadratic approximation changes from $1-x^2/2$ to -1+(x-pi)^$2/2$.

Conditions: Function is $\cos (x)$; Center is a=pi