Standard Maclaurin seriesAQA A-Level Further Maths: Flashcards

What these 13 flashcards ask

Maclaurin series for e^x?
Maclaurin series for \ln(1+x) and its range?
Maclaurin series for \sin x?
Maclaurin series for \cos x?
Series for (1+x)^n and its range?
Why does \sin x have only odd powers?
Range of validity of \ln(1-x)?
Range of validity of \ln(1+2x)?
First three terms of e^{-x}?
First two non-zero terms of \cos 2x?
First three terms of \frac{1}{1+x}?
Range of validity of a product of two series?
What happens to (1+x)^n when n is a positive integer?

Exam questions on Standard Maclaurin series

Let $f(x)=\mathrm{e}^{3x}$ .
Use the first three terms of the series, with $x=0.1$ , to estimate $\mathrm{e}^{0.3}$ to 3 decimal places.2 marks
Let $g(x)=\ln(1+2x)$ .
Use the series for $g(x)$ , and for $g(-x)$ , to find the first two non-zero terms in the series for $\ln\left(\dfrac{1+2x}{1-2x}\right)$ .2 marks
Let $f(x)=\mathrm{e}^{x}\sin x$ .
Use standard series to find the expansion of $f(x)$ in ascending powers of $x$ , up to and including the term in $x^3$ .3 marks

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).