AP Calculus BC

Taylor vs Maclaurin Series

A Maclaurin series is a Taylor series centred at zero. They are not different objects: the Maclaurin case is simply the most common centre, which is why the standard series for the exponential, sine, and cosine functions are all Maclaurin series.

Taylor series

Use when: The centre is any point, often chosen because the derivatives are easy there.

Maclaurin series

Use when: The centre is zero, which is the default unless the problem says otherwise.

Side by side

	Taylor	Maclaurin
Centre	Any $a$	Zero
Powers of	$(x - a)$	$x$
Coefficient	$\frac{f^{(n)}(a)}{n!}$	$\frac{f^{(n)}(0)}{n!}$
Relationship	The general case	The special case at $a = 0$

Choosing a centre is a practical decision. To approximate $\ln(1.1)$ you would centre at 1, because the derivatives of $\ln x$ are clean there and zero is not even in the domain.

In practice most series are built by substituting into a known Maclaurin series rather than differentiating repeatedly. The four worth memorizing are those for $e^x$ , $\sin x$ , $\cos x$ , and $\frac{1}{1-x}$ .

Frequently asked questions

Does the centre change the interval of convergence?

Yes. The interval is centred on the centre, so moving it moves the whole interval even when the radius is unchanged.

In the CED: Unit 10: Infinite Sequences and Series (BC)