The Taylor series and the Maclaurin series are both infinite sums of derivatives of a function. However, there are some distinct differences between the two series. In this blog post, we will explore those differences and see how they can be used to solve problems. We will also take a look at some examples to help illustrate these concepts.
What is The Taylor?
The Taylor sums is an approximation of a function that is calculated by taking the first few terms of a Taylor series. The Taylor series is an infinite series that is used to represent a function as a power series. The advantage of using a Taylor series is that it can be used to approximate a function at any point, not just at the point where it is defined.
- The disadvantage of using a Taylor series is that it is often difficult to calculate the coefficients in the series. The Taylor sum is one way to overcome this disadvantage by approximating the function using only the first few terms of the series. The advantage of using The Taylor sums over other methods is that it is usually more accurate than other methods, and it can be used to approximate a function at any point, not just at the point where it is defined.
- The disadvantage of using The Taylor sums over other methods is that it can be time-consuming to calculate the coefficients in the series. The Taylor sums is an approximation of a function that is calculated by taking the first few terms of a Taylor series.
- The advantage of using The Taylor sums over other methods is that it is usually more accurate than other methods, and it can be used to approximate a function at any point, not just at the point where it’s defined. The disadvantage of using The Taylor sums over other methods is that calculating the coefficients in the series can take some time.
What is Maclaurin Series?
- Maclaurin Series is a mathematical series that is used to represent a function as an infinite summation of terms. The Maclaurin Series is often used in calculus and analysis as it allows for the determination of derivatives and integrals of a function at a given point.
- The Maclaurin Series is named after Scottish mathematician Colin Maclaurin, who first derived the series in the early 18th century. Maclaurin’s theorem states that if a function can be expressed as a power series, then the function’s derivatives can be determined by taking successive derivatives of the terms in the series.
- In other words, the nth derivative of a function can be found by taking the nth derivative of each term in the Maclaurin Series and summing those derivatives. Maclaurin Series is thus a powerful tool for studying functions and their properties.
Differences between The Taylor and Maclaurin Series
The Taylor and Maclaurin Series are two ways of approximating a function. The Taylor Series is defined as f(x) = f(a) + (x – a)f'(a) + ((x – a)^2)/2! f”(a) + … while the Maclaurin Series is just the special case of the Taylor Series when a=0. So in other words, the Maclaurin Series is defined as f(x) = f(0) + xf'(0) + (x^2)/2! f”(0) + … . Many common functions can be expanded in either a Taylor or Maclaurin series.
- For example, the function e^x can be expanded in both: e^x = 1+ x+ (x^2)/2! + (x^3)/3! …. This is a Taylor series centered at x=0. And similarly: e^x = 1+ x+ (x^2)/2! + (x^3)/3! …. This is a Maclaurin series.
- As you can see, they are algebraically identical except for the fact that all the terms have been shifted over by one in the Taylor series (because we’re now centering it at x=1 instead of x=0).
- In general, a Taylor series will always have more terms than a Maclaurin series for the same function because when you center it at some point other than 0, you’ll always have at least one term that doesn’t cancel out. Higher-order derivatives will also introduce new terms that weren’t there before.
So in general, a function has an infinite number of Maclaurin series but only one Taylor series centered at any given point. That’s why people usually just refer to “the” Taylor series and “the” Maclaurin series– because really there are infinitely many of each but it’s easier to just think about them as one concept.
Conclusion
The Taylor and Maclaurin series are both important in the study of mathematics, but they have different applications. The Taylor series is more commonly used because it is able to approximate functions more accurately than the Maclaurin series. However, there are some cases where the Maclaurin series is better suited. It is important for mathematicians to be familiar with both so that they can choose the best one for each individual problem.
