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Series expansions and small-angle approximations topic check

Practise series expansions and small-angle approximations in H2 Mathematics, check independently, and return later to review. This check covers selected skills, not the whole course topic.

Learning goals

Expand by substitution into standard series and state where the standard expansions are valid
Find leading Maclaurin terms of products, quotients and compositions of standard series
Find leading Maclaurin terms of an implicitly defined function from its relation and value at zero
Use small-angle approximations to give a quadratic approximation of a trigonometric expression

GCE A-Level H2 Mathematics 2027

Start · Practise · Check · Review

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Answer each question and say how sure you are to find what to practise next.

Choose a practice setExpand standard series and state validity · 8 questionsUse small-angle approximations · 4 questionsCombine series in products and compositions · 6 questionsFind series from implicit relations · 4 questionsAll skills in this check

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Practise

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Series approximations

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Series approximations

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Series approximations

Expand standard series and state validity · Check your progress

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Series approximations

Use small-angle approximations · Check your progress

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Series approximations

Combine series in products and compositions · Check your progress

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Series approximations

Find series from implicit relations · Check your progress

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Series approximations

Check again

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Series approximations

Expand standard series and state validity · Check again

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Series approximations

Use small-angle approximations · Check again

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Series approximations

Combine series in products and compositions · Check again

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Series approximations

Find series from implicit relations · Check again

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Series approximations

Retention check

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Series approximations

Expand standard series and state validity · Retention check

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Series approximations

Use small-angle approximations · Retention check

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Series approximations

Combine series in products and compositions · Retention check

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Series approximations

Find series from implicit relations · Retention check

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Series approximations

Practise: Expand standard series and state validity

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Series approximations

Practise: Use small-angle approximations

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Series approximations

Practise: Combine series in products and compositions

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Series approximations

Practise: Find series from implicit relations

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Series approximations

Revision notes

Revise series approximations

Choose valid local expansions, derive composite and implicit coefficients, and check the accuracy of truncated approximations.

Key ideas

Standard series and convergence
Choose standard expansions with their transformed intervals, then distinguish convergence, domain and useful numerical accuracy.
Composite and implicit series
Combine series through a chosen order, derive local coefficients by repeated differentiation, and verify implicit approximations.

What the syllabus asks

From the GCE A-Level H2 Mathematics 2027 syllabus

Include: standard expansions of (1 + x)^n for rational n, e^x, sin x, cos x and ln(1 + x); first few terms by repeated differentiation (e.g. sec x), repeated implicit differentiation (e.g. y³ + y² + y = x² − 2x), or standard series (e.g. e^x cos 2x, ln((1 + x)/(1 − x))); convergence ranges; approximation of a function; small-angle approximations sin x ≈ x, cos x ≈ 1 − x²/2, tan x ≈ x. Exclude derivation of a general term.