Series approximations
Choose valid local expansions, derive composite and implicit coefficients, and check the accuracy of truncated approximations.
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Before you begin
Choose valid local expansions, derive composite and implicit coefficients, and check the accuracy of truncated approximations.
Learning goals
- Use and derive bounded Maclaurin approximations
Syllabus statements covered
- 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.
- 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.
Lessons
Work through them in order.
About lesson progress
Lesson marks record what you have studied. Checks show understanding separately. Guest marks come from this device.
- Standard series and convergenceChoose standard expansions with their transformed intervals, then distinguish convergence, domain and useful numerical accuracy.
- Composite and implicit seriesCombine series through a chosen order, derive local coefficients by repeated differentiation, and verify implicit approximations.