Taylor series and limitsEdexcel A-Level Further Maths: Mind map

Exam questions on Taylor series and limits

1. The function $f(x)=\ln x$ is expanded as a Taylor series in ascending powers of $(x-2)$.
   Use the series up to and including the term in $(x-2)^3$ to estimate $\ln2.2$, giving your answer to 4 decimal places.2 marks
2. The Maclaurin series $\mathrm{e}^{x}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots$, $\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots$ and $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots$ may be used. All limits are as $x\to0$.
   Find $\displaystyle\lim_{x\to0}\frac{\mathrm{e}^{2x^2}-1}{x^2}$.2 marks
3. Let $f(x)=\cos x$.
   Find the Taylor series of $f(x)$ in ascending powers of $\left(x-\frac{\pi}{3}\right)$, up to and including the term in $\left(x-\frac{\pi}{3}\right)^3$.3 marks