[DEFINITIONS] Taylor polynomials

Taylor polynomial

If $f$ has derivatives through order $n$ at $a$ , its Taylor polynomial of degree at most $n$ about $a$ is

P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k.

The polynomial is centred at $a$ : its powers measure displacement from $a$ , not from the origin. The factorial occurs because the $k$ -th derivative of $(x-a)^k$ at $a$ is $k!$ . In this chapter $P_n(x)$ names the Taylor polynomial; the similar notation $P_n(F)$ in vector spaces denotes the whole space of polynomials of degree at most $n$ over a field $F$ .

[1]:Taylor polynomials and a Lagrange remainder
[2]:Seen Questions
[3]:Sequences and limits
[4]:Maclaurin polynomial
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Maclaurin polynomial

The Maclaurin polynomial of degree at most $n$ is the Taylor polynomial about $a=0$ :

P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(0)}{k!}x^k.

Maclaurin polynomials are convenient when the derivatives at zero have a simple pattern.

[1]:Taylor polynomial

Definitions from Taylor polynomials

Taylor polynomial

Maclaurin polynomial

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