Revision Points - Maclaurin’s Series Expansion

Understanding Maclaurin’s Series

Take into account that a Maclaurin series expansion is a way to express a function as an infinite sum of terms calculated from the values of its derivatives at a single point.
Recognise that it’s a special case of the Taylor Series expansion when the point around which we expand is zero.
The formula for a Maclaurin series is f(x) = f(0) + f’(0)x + f’‘(0)x^2/2! + f’’‘(0)x^3/3! + …

Calculating Terms in Maclaurin’s Series

Be conscious that the nth term of a Maclaurin series is given by the nth derivative of the function evaluated at zero, divided by n!, and multiplied by x^n.
Familiarise this process by finding the Maclaurin series for familiar functions such as e^x, sin(x), cos(x), and ln(1+x).
Remember that to find the nth derivative of a function, you simply differentiate it n times.

Approximations Using Maclaurin’s Series

Keep in mind the remainder term, or error term, which estimates the error made when approximating a function using a finite number of terms from its Maclaurin series.
Know the ways to express the remainder term, including the Lagrange form of the remainder.
Be aware that the more terms are used, the more accurate the approximation of the function.

Applications of Maclaurin’s Series

Recognise the importance of Maclaurin’s series expansions in physics and engineering for simplifying calculations involving certain functions by approximating them using a polynomial.
Discover the numerous applications in analytical and numerical calculations, including the evaluation of definite integrals, numerical analysis and solving differential equations.

Repetition is key in mastering Maclaurin’s Series Expansion. Regular practice in finding series for different functions and in different contexts will improve familiarity and understanding of this concept. This topic is a bedrock of many parts of Further Maths.