Maclaurin series of e^x?

1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots (all x).

Maclaurin series of \sin x?

x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots (all x).

Maclaurin series of \cos x?

1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots (all x).

Maclaurin series of \ln(1+x) and its validity?

x-\frac{x^2}{2}+\frac{x^3}{3}-\dots for -1<x\le1.

-x-\frac{x^2}{2}-\frac{x^3}{3}-\dots

Series of \tan x up to x^5?

x+\frac{x^3}{3}+\frac{2x^5}{15}.

How do you find \frac{d^3y}{dx^3} for y=\tan x quickly?

Use y'=1+y^2: y''=2yy', y'''=2y'^2+2yy''.

How do you find a limit as x\to0 using series?

Replace each function by its series, cancel the leading power of x, then let x\to0.

Maclaurin seriesEdexcel International A Level Further Maths: Flashcards

Card 1 of 130 of 13 known

Question

State the Maclaurin series of $f(x)$.

Tap or press Space to reveal

Tap card or press Space to flip

See all 13 cards

State the Maclaurin series of $f(x)$ .: $f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\frac{x^3}{3!}f'''(0)+\dots+\frac{x^n}{n!}f^{(n)}(0)+\dots$
Maclaurin series of $\mathrm{e}^x$ ?: $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots$ (all $x$ ).
Maclaurin series of $\sin x$ ?: $x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots$ (all $x$ ).
Maclaurin series of $\cos x$ ?: $1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots$ (all $x$ ).
Maclaurin series of $\ln(1+x)$ and its validity?: $x-\frac{x^2}{2}+\frac{x^3}{3}-\dots$ for $-1<x\le1$ .
Series of $\ln(1-x)$ ?: $-x-\frac{x^2}{2}-\frac{x^3}{3}-\dots$
Series of $\mathrm{e}^{2x}$ up to $x^3$ ?: $1+2x+2x^2+\frac43x^3$ .
Series of $\ln\frac{1+x}{1-x}$ up to $x^3$ ?: $2x+\frac23x^3$ .
Series of $\tan x$ up to $x^5$ ?: $x+\frac{x^3}{3}+\frac{2x^5}{15}$ .
Series of $\cos2x$ up to $x^4$ ?: $1-2x^2+\frac23x^4$ .
How do you find $\frac{d^3y}{dx^3}$ for $y=\tan x$ quickly?: Use $y'=1+y^2$ : $y''=2yy'$ , $y'''=2y'^2+2yy''$ .
What is $f^{(n)}(0)$ in terms of the series coefficient $a_n$ ?: $f^{(n)}(0)=n!\,a_n$ .
How do you find a limit as $x\to0$ using series?: Replace each function by its series, cancel the leading power of $x$ , then let $x\to0$ .

Exam questions on Maclaurin series

A function is defined by $f(x)=\mathrm{e}^{3x}$ .
Write down the Maclaurin series of $f(x)$ up to and including the term in $x^3$ , and use it with $x=0.3$ to estimate $\mathrm{e}^{0.9}$ to 3 significant figures.2 marks
A function is defined by $f(x)=\cos2x$ .
Use the identity $\cos2x=1-2\sin^2x$ and the Maclaurin series of $\cos2x$ up to the term in $x^4$ to find the first two non-zero terms of the Maclaurin series of $\sin^2x$ .2 marks
A function is defined by $f(x)=\ln(1+x)$ , for $-1<x\le1$ .
Use differentiation to find the Maclaurin series of $f(x)$ up to and including the term in $x^3$ .3 marks

See the full worksheet

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).