Series (FP2) Topic Review summary

Key learning points

• Maclaurin: $f(x)=f(0)+f'(0)x+\\frac{f''(0)}{2!}x^2+\\cdots$.
• Standard expansions for $e^x$, $\\sin x$, $\\cos x$, $\\ln(1+x)$, $(1+x)^n$.
• Differentiate/integrate series within the radius of convergence.
• Approximate values with a stated number of terms and error discussion if required.
• Combine with complex form $e^{i\\theta}$ when the course links them.

Worked example 1 — e^x

Maclaurin for $e^x$ up to $x^3$.

Working

All derivatives are $e^x$.

Worked example 2 — ln

$\\ln(1+x)$ up to $x^3$.

Working

Valid for $-1<x\\le1$.

Worked example 3 — Practice point 1

Maclaurin: $f(x)=f(0)+f'(0)x+\\\\frac{f''(0)}{2!}x^2+\\\\cdots$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Practice point 2

Standard expansions for $e^x$, $\\\\sin x$, $\\\\cos x$, $\\\\ln(1+x)$, $(1+x)^n$.

Working

Attempt without notes, then check the answers PDF.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.