Taylor Polynomials

## Description

The $n$-degree Taylor polynomial of a function $f(x)$ at a point $x = x_0$ is defined as

(1)\begin{align} T_n(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{f''(x_0)}{2!}(x - x_0)^2 + \cdots + \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n. \end{align}

We can use Sage to compute Taylor polynomials. As an example, we can compute the 8-th degree Taylor polynomial of $f(x) = \cos x$ at the point $x_0 = 0$.

## Sage Cell

#### Code

```
f(x) = cos(x)
T(x) = taylor( f(x), x, 0, 8 )
T
```

## Options

#### Option

None

## Tags

Primary Tags: Calculus - Single Variable

Secondary Tags: Infinite Sequences and Series: Taylor Series

## Related Cells

Related cells: None

## Attribute

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Author: T. W. Judson

Date: 15 May 2018 20:36

Submitted by: Tom Judson