Taylor Series

The expression for the order of error given above is found by comparison of the procedure with a Taylor series.

• A Taylor series is a power series that can approximate a function, f(x), for values near to x=a.
• Its coefficients use the derivatives of $f$ at $x=a$
• $f(x)=f(a)+\frac{f^{'}(a)}{1!}(x-a)+\frac{f^{''}(a)}{2!}(x-a)^2+\frac{f^{'''}(a)}{3!}(x-a)^3+\ldots$
• The Taylor series says that if we know the values for all derivatives of $f(x)$ at $x=a$, we can approximate the function as closely as we desire.
• $Error~of~TS=\frac{f^{n+1}(\xi)}{(n+1)!}$: The error term for a truncated Taylor series after the $n^{th}$ term
• where $\xi$ is a value between $x$ and $(x+a)$. Since the value of $\xi$ is not known, there is still uncertainty in the exact value of the error.

Example m-file: Taylor Series Approximations to $1/(1-x)$ (http://siber.cankaya.edu.tr/ozdogan/NumericalComputations//mfiles/chapter0/demoTaylor.m demoTaylor.m)
Consider the function

$$f(x)=\frac{1}{1-x}$$

Make the Taylor series expansion of this function up to third order.

demoTaylor(1.6,0.8)

All of the Taylor polynomials agree with $f(x)$ near $x_0=1.6$. The higher order polynomials agree over a larger range of $x$.