Question

What is the difference between a Taylor series and a MacLaurin series?

Answer 1

Hint: In above question, we are given two types of series expansion of functions, the Taylor series and the MacLaurin series. We have to determine the difference between the two series. However, both the series expansions are very similar to each other. That is in fact, the MacLaurin series is actually nothing else but a special case of the Taylor series. Let us see how.

Complete step-by-step answer:
Given series are, Taylor series and the MacLaurin series.
We have to differentiate between the two series expansions.
$\bullet$ Let us see the Taylor series first.
A Taylor series is a series expansion of a function about a known point. A one-dimensional Taylor series is an expansion of a real function $$f\left(x\right)$$ about a point $$x=a$$ is given by as follows:
$$\Rightarrow f\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{f^n\left(a\right)}{n!}}{\left(x-a\right)}^n$$
On expansion, we get the function
The above expansion is known as the Taylor series.

$\bullet$ Whereas, the MacLaurin series is a series expansion of a function about the fixed known point $$0$$ . A one-dimensional MacLaurin series is an expansion of a real function $$f\left(x\right)$$ about a point $$x=0$$ is given by as follows:
$$\Rightarrow f\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{f^n\left(0\right)}{n!}}{\left(x\right)}^n$$
On expansion, we get the function

The above expansion is known as the MacLaurin series.
Now, we can also write the MacLaurin series as,

We can notice that above expansion is in fact a Taylor series when $$a=0$$ .
Therefore the MacLaurin series is nothing but a special case of the Taylor series when $$a=0$$ .

Note: There is a required condition for a function to be expandable through the Taylor and Maclaurin series, that is the function must be continuous as well as differentiable in the range of real numbers. The Taylor and Maclaurin series can be used to calculate the value of a whole function at every point, if the value of the function, and of all of its derivatives, are known at a single point $$a$$ or $$0$$ . The partial sums (Taylor polynomials) of the series can be used as approximations of the function.