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LearnEdexcelYFM012018 specificationWFM02 · Further Pure Mathematics 2Maclaurin and Taylor series
WFM02 · Module

Maclaurin and Taylor series

2 subtopics≈ 95 min in total
Start the module: Maclaurin seriesOpen graphing calculator

  1. Maclaurin series

    Any smooth function can be approximated near 00 by a polynomial whose derivatives match it there: f(x)=f(0)+f′(0)x+f′′(0)2!x2+⋯f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\cdots. Standard series for exe^x, sin⁡x\sin x, cos⁡x\cos x, ln⁡(1+x)\ln(1+x) and (1+x)n(1+x)^n combine by substitution, multiplication and differentiation.

    ≈ 45 min

  2. Taylor series, L'Hôpital and Leibniz

    Expanding about x=ax=a instead of 00 gives the Taylor series f(x)=∑f(r)(a)r!(x−a)rf(x)=\sum\frac{f^{(r)}(a)}{r!}(x-a)^r. Series also crack "00\frac00" limits (as does L'Hôpital's rule), and Leibniz's theorem gives the nnth derivative of a product like a binomial expansion.

    ≈ 50 min