What is the difference between the Taylor and Maclaurin series?

Question

What is the difference between the Taylor and the Maclaurin series? Is the series representing sine the same both ways? Can someone describe an example for both?

Answer 1

A Taylor series centered at $x=x_0$ is given as follows:

$$f(x)=\sum_{n=0}^\infty\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$

while a Maclaurin series is the special case of being centered at $x=0$:

$$f(x)=\sum_{n=0}^\infty\frac{f^{(n)}(0)}{n!}x^n$$

You may find this very similar to a power series, which is of the form

$$f(x)=\sum_{n=0}^\infty a_n(x-x_0)^n$$

Particularly where $a_n=\frac{f^{(n)}(x_0)}{n!}$. If a function is equal to it's Taylor series locally, it is said to be an analytic function, and it has a lot of interesting properties. However, not all functions are equal to their Taylor series, if a Taylor series exists.

One may note that most of the most famous Taylor series are a Maclaurin series, probably since they look nicer. For example,

$$\sin(x)=\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!}$$

or,

$$\sin(x)=\sum_{n=0}^\infty\frac{(-1)^n(x-2\pi)^{2n+1}}{(2n+1)!}$$

Which is trivially due to the fact that $\sin$ is a periodic function. So, if you had to choose, you'd probably choose the first representation. Just a convention.

The geometric series is a rather beautifully known Maclaurin series, which one may derive algebraically without taking derivatives:

$$\frac1{1-x}=\sum_{n=0}^\infty x^n=1+x+x^2+x^3+\dots$$

However, it gets a little bit more involved when you try to take the Taylor series at a different point.

Answer 2

A MacLaurin series is a special occurrence of the Taylor Series where the series is constructed around x=0.

MacLaurin series are generally used if able to.

For example, you can estimate $f(x)=\sin{x}$ with a Maclaurin series.

However, you can't estimate $f(x) = \frac{1}{x}$ with a Maclaurin series because $f(x)$ is undefined when $x=0$, so most people choose to center it around $x=1$. Usage is all about preference.

Answer 3

The Taylor series is a series of functions of the form:

$$f(x)=\sum_{n=0}^{\infty}a_{n}(x-a)^n,$$ where $a_n=\frac{f^{(n)}(a)}{n!}.$ This series is called the Taylor series of $ f (x) $ around the point $ x = a. $ In the particular case of $ a = 0 $ the series is called the Maclaurin series.