Difference between Taylor and MacLaurin Series-Usage

In summary, Taylor and Maclaurin series are essentially the same, as the Taylor series about a of f(x) is equivalent to the Maclaurin series of f(x+a). However, it is better to use the series about a specific value (such as 2) when approximating a point close to that value (such as 2.1). It is important to note that when discussing accuracy, we are actually referring to Taylor and Maclaurin polynomials, not the series themselves.
  • #1
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Hi all,

I understand the numerical difference between a Taylor and Maclaurin Series; Maclaurin series is just Taylor Series about x=0. However, is there any difference between their usage?

I'm guessing Taylor series may be more accurate with less terms for approximating something close to its center; ex. finding f(2.1) for Taylor series about x=2 rather than using Maclaurin series. Am I correct in this assumption? What makes using one better than using the other?

Thanks.
 
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  • #2
Yes it may be better to use the series about 2 to find f(2.1) than the series about 0. We can do this with the Maclaurin Series though just find the series about 0 of f(x+2)

Since the Taylor series about a of f(x) is the same as the Maclaurin series of f(x+a) the two can be considered the same.
 
  • #3
A word about terminology: you are really talking about Taylor and MacLaurin polynomials (truncated series) not the series themselves. There is no question of "accuracy" with Taylor and MacLaurin series.
 

What is the difference between Taylor and MacLaurin series?

The main difference between Taylor and MacLaurin series is that Taylor series is a generalized form of MacLaurin series, which is a type of power series expansion used to represent a function as an infinite sum of terms. MacLaurin series is a special case of Taylor series, where the expansion is centered at x=0.

What is the usage of Taylor series?

Taylor series is used in calculus and mathematical analysis to approximate a function using a polynomial. It allows us to represent a complex function as a simpler polynomial, making it easier to analyze and manipulate mathematically. Taylor series can also be used to find the derivatives and integrals of a function.

What is the usage of MacLaurin series?

MacLaurin series is used to approximate a function as a polynomial, specifically when the function is centered at x=0. It is a special case of Taylor series and is useful for simplifying complex functions and making them easier to work with mathematically.

What are the similarities between Taylor and MacLaurin series?

The main similarity between Taylor and MacLaurin series is that they both use a polynomial to approximate a function. They also both involve taking derivatives of the function at a specific point to find the coefficients of the polynomial. Additionally, they both have an infinite number of terms, making them more accurate as the number of terms increases.

What are the limitations of Taylor and MacLaurin series?

The limitations of Taylor and MacLaurin series are that they can only approximate a function within a certain radius of convergence. If the function has singularities or is undefined within this radius, the series will not accurately represent the function. Additionally, the accuracy of the series depends on the smoothness of the function and the number of terms used in the expansion.

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