Practice section 4 of 7

Expand $(2+1/x)^{-2}$ in ascending powers of $x$, up to and including $x^5$. State the range of $x$ for which the expansion is valid.

The coefficient of $x^2$ in the series expansion of $\dfrac1{e^x(1+ax)}$ is $1/4$. Find the constant $a$. You may use standard results from MF27.

(i) Expand $\sqrt[3]{8+x}$ through $x^2$ and state the range of validity.

(ii) Put $x=1/27$ to approximate $\sqrt[3]{217}$ as $p/q$ in lowest terms, with $p,q\in\mathbb Z^+$.

(iii) We can also substitute $x=1/217$ into the expansion found in (i) to get an approximate value of $\sqrt[3]{217}$. Explain whether this approximation is better than the approximation in (ii).

Let $y=\ln(1+\sin x)$.

(i) Show that $e^y\dfrac{d^2y}{dx^2}=1-e^y[(\dfrac{dy}{dx})^2+1]$.

(ii) Find its Maclaurin series through $x^2$.

(iii) Verify it using the standard series for $\sin x$ and $\ln(1+x)$.

Given $y=\tan(\pi/4+x)$, show that $\dfrac{d^2y}{dx^2}=2y\dfrac{dy}{dx}$. Find its series through $x^4$. When $x=0.1$, $y=1.22304888$ correct to 8 decimal places. Verify that the fourth-degree estimate has error about $0.004\%$.

The curve $y=f(x)$ passes through $(0,\pi/4)$ and satisfies $y^\prime\tan y=\ln((e^x+1)/2)$. Find the first three nonzero Maclaurin terms of $y$. Deduce the expansion of $(\cot y)\ln((e^x+1)/2)$ through $x^2$.