# Taylor Polynomials

The $n$-th order Taylor polynomial centered at $x_{0}$ is given by $P_{n,x_{0}}(x)=∑_{k=0}k!f_{(k)}(x_{0}) (x−x_{0})_{k}$

1st order expansion $P(x)=f(x_{0})+f_{′}(x_{0})(x−x_{0})$ 2nd-order expansion $P(x)=f(x_{0})+f_{′}(x_{0})(x−x_{0})+21 f_{′′}(x_{0})(x−x_{0})_{2}$ etc.

### Shortcuts

### Maclaurin Polynomial

A Taylor polynomial centered at zero ($x_{0}=0$) is referred to as a Maclaurin polynomial. $P_{n,0}(x)=∑_{k=0}k!f_{(k)}(0) x_{k}$

### Taylor’s Remainder Theorem

Not that needed. Used to prove Taylor’s Inequality. Suppose that $f$ has $n+1$ derivatives at $x_{0}$. Then

$f(x)=∑_{k=0}k!f_{(k)}(x_{0}) (x−x_{0})_{k}+R_{n}(x)$ where $R_{n}(x)=∫_{x_{0}}n!(x−t)_{n} f_{(n+1)}(t)dt$

### 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)∣≤K(n+1)!∣x−x_{0}∣_{n+1} $ where $∣f_{(n+1)}(z)∣≤K$ for all values of $z$ between $x_{0}$ and $x$.

Note

We can use this value to bound the actual values.

$P_{n,x_{o}}(x)−R_{n}(x)≤f(x)≤P_{n,x_{0}}+R_{n}(x)$

### 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. $sinx=x−3!x_{3} +5!x_{5} −7!x_{7} +⋯$ $cosx=1−2!x_{2} +4!x_{4} −6!x_{6} +⋯$

memory trick for Even and Odd Function

WAIT,#serendipity, is this a coincidence? $sinx$ happens to be an even function, and $cosx$ happens to be an odd function.

If you look at the sums, $sinx$ uses odd powers, while $cosx$ uses even powers.

$e_{x}=1+x+2x_{2} +3!x_{3} +⋯$

### Multivariate case

In the multivariate case, $f_{′}(x)$ is given by the Jacobian, and $f_{′′}(x)$ is given by the Hessian.

I saw this when learning the Gauss-Newton Method.

$f(x+Δx)≈f(x)+J(x)_{T}Δx$