Maclaurin and Taylor Series Topic Review summary

Key learning points

• Maclaurin is Taylor about 0: use successive derivatives at the expansion point.
• Memorise core series and build others by substitution, differentiation, integration.
• State the interval/radius of convergence when infinite.
• Truncation error discussions appear in some papers — know the idea.
• Products/compositions of series for more advanced expansions.

Worked example 1 — sin

Maclaurin for $\\sin x$ up to $x^3$.

Working

Odd function — even powers vanish.

Worked example 2 — Substitute

Series for $e^{2x}$ from $e^x$.

Working

Replace $x$ by $2x$.

Worked example 3 — Practice point 1

Apply this idea from the Topic Review: Maclaurin is Taylor about 0: use successive derivatives at the expansion point.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Practice point 2

Apply this idea from the Topic Review: Memorise core series and build others by substitution, differentiation, integration.

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.