Maclaurin Series — Syllabus Points

1. Standard Maclaurin Expansions

• 1.1 Derive Maclaurin series for standard functions from first principles
• 1.2 Know the standard series for $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $\tan^{-1}x$
• 1.3 Use the general formula $f(x)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$

2. Composite Functions

• 2.1 Substitute into standard series to expand composite functions ($e^{f(x)}$, $\ln(g(x))$)
• 2.2 Use logarithmic differentiation before expansion where appropriate
• 2.3 Multiply/divide series to obtain required terms

3. Series Involving Powers

• 3.1 Expand functions like $a^x = e^{x\ln a}$ via Maclaurin series
• 3.2 Handle functions with negative/fractional powers using binomial series

4. Approximation of Integrals

• 4.1 Integrate series term-by-term to approximate definite integrals
• 4.2 Determine the number of terms needed for a given accuracy

5. Error and Validity

• 5.1 State the range of validity for Maclaurin expansions where required
• 5.2 Understand that series expansions are approximations when truncated