State the Taylor series in terms of h where x=a+h.

f(a+h)=f(a)+hf'(a)+\frac{h^2}{2!}f''(a)+\dots

How is the Maclaurin series related to the Taylor series?

It is the Taylor series with a=0.

Taylor series of \ln x about x=2 up to (x-2)^3?

\ln2+\frac12(x-2)-\frac18(x-2)^2+\frac1{24}(x-2)^3.

Taylor series of \sqrt x about x=4 up to (x-4)^2?

2+\frac14(x-4)-\frac1{64}(x-4)^2.

Derivatives of \tan x: first three?

\sec^2x, 2\sec^2x\tan x, 4\sec^2x\tan^2x+2\sec^4x.

How do you choose the centre a when approximating f(x0)?

Pick a close to x0 with exact, simple values of f(a) and its derivatives.

Quick check for a Taylor series of \sin x about \pi?

Put x=\pi+h and use \sin(\pi+h)=-\sin h.

Taylor seriesEdexcel International A Level Further Maths: Flashcards

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State the Taylor series of $f(x)$ about $x=a$.

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State the Taylor series of $f(x)$ about $x=a$ .: $f(a)+(x-a)f'(a)+\frac{(x-a)^2}{2!}f''(a)+\frac{(x-a)^3}{3!}f'''(a)+\dots$
State the Taylor series in terms of $h$ where $x=a+h$ .: $f(a+h)=f(a)+hf'(a)+\frac{h^2}{2!}f''(a)+\dots$
How is the Maclaurin series related to the Taylor series?: It is the Taylor series with $a=0$ .
What is the coefficient of $(x-a)^n$ in a Taylor series?: $\frac{f^{(n)}(a)}{n!}$ .
Taylor series of $\sin x$ about $\pi$ up to $(x-\pi)^3$ ?: $-(x-\pi)+\frac{(x-\pi)^3}{6}$ .
Taylor series of $\ln x$ about $x=2$ up to $(x-2)^3$ ?: $\ln2+\frac12(x-2)-\frac18(x-2)^2+\frac1{24}(x-2)^3$ .
Taylor series of $\sqrt x$ about $x=4$ up to $(x-4)^2$ ?: $2+\frac14(x-4)-\frac1{64}(x-4)^2$ .
Taylor series of $\tan x$ about $\frac{\pi}{4}$ up to $h^3$ ( $h=x-\frac{\pi}{4}$ )?: $1+2h+2h^2+\frac83h^3$ .
Derivatives of $\tan x$ : first three?: $\sec^2x$ , $2\sec^2x\tan x$ , $4\sec^2x\tan^2x+2\sec^4x$ .
How do you choose the centre $a$ when approximating $f(x_0)$ ?: Pick $a$ close to $x_0$ with exact, simple values of $f(a)$ and its derivatives.
Where are the derivatives evaluated in a Taylor series?: At the centre $x=a$ , not at $0$ .
Quick check for a Taylor series of $\sin x$ about $\pi$ ?: Put $x=\pi+h$ and use $\sin(\pi+h)=-\sin h$ .

Exam questions on Taylor series

A function is defined by $f(x)=\ln x$ , for $x>0$ . Its Taylor series about $x=2$ is to be found.
Use the Taylor series up to and including the term in $(x-2)^3$ to estimate $\ln2.1$ to 5 significant figures, given that $\ln2=0.693147$ to 6 significant figures.2 marks
A function is defined by $f(x)=\sqrt{x}$ , for $x>0$ . Its Taylor series about $x=4$ is to be found.
The Taylor series about $x=4$ up to the term in $(x-4)^3$ is $2+\frac14(x-4)-\frac{1}{64}(x-4)^2+\frac{1}{512}(x-4)^3$ . Use it to estimate $\sqrt{4.2}$ to 4 decimal places.2 marks
A function is defined by $f(x)=\sin x$ . Its Taylor series about $x=\pi$ is to be found.
Find the Taylor series of $f(x)$ in ascending powers of $(x-\pi)$ up to and including the term in $(x-\pi)^3$ .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).