Exam Questions - Maclaurin’s series

Maclaurin’s Series Basics

The Maclaurin’s Series is a Taylor series expansion of a function about 0. In essence, it provides a way to predict a function’s behaviour based on its derivatives at a single point.
The formula for a Maclaurin’s series is f(x) = f(0) + f’(0)x + f’‘(0)x^2/2! + f’’‘(0)x^3/3! + …
The notation f’(0), f’‘(0), f’’‘(0), etc., refer to the first, second, third, etc., derivatives of the function evaluated at x = 0.
The “!” symbol signifies a factorial, the product of an integer and all the integers below it.

Generating Maclaurin’s Series

For most questions involving Maclaurin’s Series, you will need to generate the series for a given function. This involves calculating the function’s derivatives at x = 0 and substituting them into the series formula.
For example, if given the function f(x) = e^x, we know that the derivative of e^x is e^x for all orders, so f(0), f’(0), f’‘(0) are all 1. This leads to the Maclaurin series: f(x) = 1 + x + x^2/2! + x^3/3! + …
Note that some functions have known Maclaurin series. For instance, for sin(x) it’s x - x^3/3! + x^5/5! - x^7/7! +…

Approximations Using Maclaurin’s Series

Maclaurin’s series can be used to approximate values of a function near x = 0. The more terms you include, the better the approximation.
To use the series as an approximation, determine the number of terms the question requires. Plug the necessary x-value into those terms and calculate.
Be aware that even when using many terms, an approximation is still an approximation, and will not be exactly equal to the function’s true value. However, for many practical purposes, the approximation can be considered close enough.

Error Bound in Maclaurin’s Series

The difference between the true value of the function and the approximation obtained by truncating the series is called the remainder or error.
In questions involving error estimation, the Lagrange form of the remainder is commonly employed: Rn = f^(n+1)(c)x^(n+1)/(n+1)! where c is a number between 0 and x.
Understanding and being able to apply this form can provide a maximum bound for the error, shedding light on the accuracy of the approximation.

Manipulating Maclaurin’s Series

Often problems will require manipulation of a known Maclaurin’s Series to find a new series. For instance, you might need to find the series for 2e^x, which can be obtained by multiplying every term in the series for e^x by 2.
When integrating or differentiating a Maclaurin’s series, apply the operation term-by-term to each term of the series. For example, to find the series for e^x^2, consider squaring every term in the series for e^x.

Remember, practice is key when understanding and mastering Maclaurin’s series, so regular revision of the rules and extensive practice of past questions are essentials to excel in this topic.