Edexcel IAL - Further Pure Mathematics 2- 6.2 Maclaurin Series- Study notes - New syllabus
Edexcel IAL – Further Pure Mathematics 2- 6.2 Maclaurin Series -Study notes- New syllabus
Edexcel IAL – Further Pure Mathematics 2- 6.2 Maclaurin Series -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 6.2 Maclaurin Series
Maclaurin Series
The Maclaurin series is a special case of the Taylor series, where the expansion is taken about \( x = 0 \). It allows complicated functions to be approximated by polynomials.
If a function \( f(x) \) has derivatives of all orders at \( x = 0 \), then it can be expanded as
\( f(x) = f(0) + f'(0)x + \dfrac{f”(0)}{2!}x^2 + \dfrac{f”'(0)}{3!}x^3 + \cdots \)
Standard Maclaurin Series
| Function | Maclaurin Series |
| \( e^x \) | \( 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots \) |
| \( \sin x \) | \( x – \dfrac{x^3}{3!} + \dfrac{x^5}{5!} – \cdots \) |
| \( \cos x \) | \( 1 – \dfrac{x^2}{2!} + \dfrac{x^4}{4!} – \cdots \) |
| \( \ln(1+x) \) | \( x – \dfrac{x^2}{2} + \dfrac{x^3}{3} – \cdots \) |
Why Maclaurin Series Are Useful
- To approximate functions for small values of \( x \).
- To evaluate limits.
- To simplify complicated expressions.
- To solve differential equations approximately.
Example
Write the Maclaurin expansion of \( e^x \) up to the term in \( x^3 \).
▶️ Answer / Explanation
\( e^x = 1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{6} \)
Example
Find the Maclaurin series for \( \sin x \) up to \( x^5 \).
▶️ Answer / Explanation
\( \sin x = x – \dfrac{x^3}{6} + \dfrac{x^5}{120} \)
Example
Use a Maclaurin series to approximate \( \ln(1.1) \).
▶️ Answer / Explanation
\( \ln(1+x) \approx x – \dfrac{x^2}{2} + \dfrac{x^3}{3} \)
With \( x = 0.1 \):
\( \ln(1.1) \approx 0.1 – 0.005 + 0.000333 = 0.09533 \)