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Maclaurin series of a functionAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

  • State Maclaurin's theorem.
  • What is the coefficient of x^r in a Maclaurin series?
  • How do you find f^{(r)}(0) from a known series?
  • Maclaurin series of \tan x (to x^3)?
  • First two non-zero terms of \ln\cos x?
  • General term of \frac{1}{1-x}?
  • General term of e^{2x}?
  • General term of \ln(1+x)?
  • General term of \sin x?
  • If f is an odd function, which powers of x appear?
  • Why can't \ln x have a Maclaurin series?
  • Maclaurin series of e^{\sin x} to x^3?

Exam questions on Maclaurin series of a function

  1. Let f(x)=ln⁡(cos⁡x)f(x)=\ln(\cos x) for −π2<x<π2-\frac{\pi}{2}<x<\frac{\pi}{2}.
    Given that f′′′(0)=0f'''(0)=0 and f(4)(0)=−2f^{(4)}(0)=-2, find the first two non-zero terms of the Maclaurin series of f(x)f(x).2 marks
  2. Let f(x)=11−2xf(x)=\dfrac{1}{1-2x}.
    Deduce an expression for f(r)(0)f^{(r)}(0) in terms of rr.2 marks
  3. Let f(x)=tan⁡xf(x)=\tan x.
    Show that f′′′(0)=2f'''(0)=2.3 marks
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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).