Taylor Polynomials

The $n$-th order Taylor polynomial centered at $x_0$ is given by $P_{n,x_0}(x)={\sum }_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0{)}^k$

1st order expansion $P(x)=f(x_0)+f^{\prime }(x_0)(x-x_0)$ 2nd-order expansion $P(x)=f(x_0)+f^{\prime }(x_0)(x-x_0)+\frac{1}{2}{f}^{\prime \prime }(x_0)(x-x_0{)}^2$ etc.

Shortcuts

Maclaurin Polynomial

A Taylor polynomial centered at zero ($x_0=0$) is referred to as a Maclaurin polynomial. $P_{n,0}(x)={\sum }_{k=0}^n\frac{f^{(k)}(0)}{k!}{x}^k$

Taylor’s Remainder Theorem

Not that needed. Used to prove Taylor’s Inequality. Suppose that $f$ has $n+1$ derivatives at $x_0$. Then

$f(x)={\sum }_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0{)}^k+R_n(x)$ where $R_n(x)={\int }_{x_0}^x\frac{(x-t{)}^n}{n!}{f}^{(n+1)}(t)dt$

Taylor’s Inequality

The error in using an nth-order Taylor polynomial $P_{n,x_0}(x)$ as an approximation to $f(x)$ satisfies the inequality $|R_n(x)|\le K\frac{|x-x_0{|}^{n+1}}{(n+1)!}$ where $|f^{(n+1)}(z)|\le K$ for all values of $z$ between $x_0$ and $x$.

Note

We can use this value to bound the actual values.

$P_{n,x_o}(x)-R_n(x)\le f(x)\le P_{n,x_0}+R_n(x)$

Approximation of Integrals using Taylor Polynomials

Note

If you want tighter bounds, usually you can increase the degree of the polynomial and that will allow you to have tigher ranges.

Taylor Series (Expanded)

We can use taylor series to approximate functions. See Power Series for non-expanded version. $\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$ $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$

memory trick for Even and Odd Function

WAIT,#serendipity, is this a coincidence? $\sin x$ happens to be an even function, and $\cos x$ happens to be an odd function.

If you look at the sums, $\sin x$ uses odd powers, while $\cos x$ uses even powers.

$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\cdots$

Multivariate case

In the multivariate case, $f^{\prime }(x)$ is given by the Jacobian, and $f^{\prime \prime }(x)$ is given by the Hessian.

I saw this when learning the Gauss-Newton Method.

$f(x+\Delta x)\approx f(x)+J(x{)}^T\Delta x$

Related

• Series
• Power Series