Maclaurin and Taylor Series summary

Key learning points

• $f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\cdots$.
• Memorise standard expansions for $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$.
• Differentiate/integrate series inside the radius of validity.
• State validity ranges when required.

Worked example 1 — e^x

First four Maclaurin terms for $e^x$.

Working

Factorials appear in the denominators.

Worked example 2 — Key skill 1

Use this method from the Maclaurin and Taylor Series sheet: $f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\cdots$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 3 — Memorise standard expansions for , , ,

Use this method from the Maclaurin and Taylor Series sheet: Memorise standard expansions for $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 4 — Differentiate/integrate series inside the radius

Use this method from the Maclaurin and Taylor Series sheet: Differentiate/integrate series inside the radius of validity.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 5 — State validity ranges when required

Use this method from the Maclaurin and Taylor Series sheet: State validity ranges when required.

Working

Each key learning point matches a question type on the printable worksheet.

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.