Review practice · Section 1 of 2

Express $f(x)=\dfrac{3x^2-3x+4}{(2x-1)(x^2+3)}$ in partial fractions and hence obtain $f(x)$ as a series expansion in ascending powers of $x$ up to and including the term in $x^2$.5 marks

Find the range of values of $x$ for which the expansion is valid.1 mark

Expand $(4-x)^{-1/2}$ in ascending powers of $x$ through $x^3$, and state its range of validity. Find $a$ if the term independent of $x$ in $(1+ax^{-1})/\sqrt{4-x}$ is $5$.6 marks

Expand $\sqrt{1-x}/(1+2x)$ through $x^2$.4 marks

(i) A student substitutes $x=-1$ to estimate $\sqrt2$. Explain why this is wrong.1 mark

(ii) Put $x=1/9$ in the expansion to show $\sqrt2\approx p/1296$, and find the positive integer $p$.2 marks

Given $y=e^{\cos x}$, show $y^{\prime\prime}=-y\cos x-y^{\prime}\sin x$. Show that the first two nonzero terms in its expansion are $e-\tfrac12ex^2$.4 marks

For $y=e^x\sin x$, show $y^{\prime\prime}=2(y^{\prime}-y)$. Differentiate this further to find the Maclaurin series through $x^3$, then verify it using the standard series for $e^x$ and $\sin x$.7 marks

Given $y=\ln(1+e^x)$, show $y^{\prime\prime}=y^{\prime}-(y^{\prime})^2$.3 marks

Hence find its Maclaurin series through $x^2$.2 marks

Deduce the first two nonzero terms in the expansion of $e^x/(1+e^x)$.3 marks