`taylor`

Taylor series expansion of a symbolic expression about a point.

Syntax

T = taylor(f, x)

Expands f in powers of x about 0, keeping terms below degree 6.

T = taylor(f, x, a)

Expands about the point a.

T = taylor(___, Name, Value)

Sets 'Order' or 'ExpansionPoint' by name.

Description

taylor builds the truncated series f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... by differentiating f symbolically and evaluating the derivatives at the expansion point. The remainder term is not returned.

The Order option counts terms, not the degree of the last term, so 'Order', 4 stops at the cubic term.

Input arguments

f: Symbolic expression or symbolic function.
x: Symbolic variable to expand in.
'Order': Number of terms: the result has degree at most Order - 1. Default 6.
'ExpansionPoint': Point a the series is centered on. Default 0.

Output arguments

T: Symbolic polynomial in x (or in x - a).

Example

Maclaurin and shifted series

syms x
taylor(exp(x), x, 'Order', 4)
taylor(cos(x), x, 'ExpansionPoint', 1, 'Order', 3)

Output

ans = x^3/6 + x^2/2 + x + 1
ans = cos(1) - (x - 1)*sin(1) - ((x - 1)^2*cos(1))/2

taylor