> See also: > - [[Sums and Series#Power Series]] # Taylor Series and Polynomials If $f$ is a function defined on an interval $(c,b)$ $f(x)=f(a)+(x-a)\cdot f'()$ --- The formula for the Taylor Series expansion of a function $f(x)$ is: $\sum^{\infty}_{n=0} \frac{(x-a)^n}{n!} \cdot f^{(n)}(a)$ The variable $a$ is known as the *point of expansion*. A Taylor series can also be described as “$f$ about $a$” or “$f$ centered at $a$”. A Taylor series is a type of power series where the coefficients are instead defined in terms of the derivatives of the function. ## Taylor Polynomials ![[Pasted image 20240418070243.png|400]] $P_n(x)=f(a)+(x-a)f'(a)$ $f(x) = P_n(x) + R_n(x)$ The Taylor Polynomial can be considered a *partial sum* of the full Taylor Series of a function As the degree of the Taylor polynomial increases, so does the accuracy of the approximation of the function around the point of expansion. For some functions (Ex: polynomials), their higher order derivatives will eventually become 0 The Taylor remainder *quantifies the difference* between the function being approximated and its Taylor polynomial approximation. - ### Maclaurin Series The Maclaurin series is the Taylor series when the point of expansion is set to 0. $P_0(x)=f(0)$ ## Applications of Taylor Series When you evaluate a Taylor series expansion at a specific value of $x$, you will get an approximation of the original function at that point. ### Analysis of Complex Functions