2021 TMJC P2 Q4
(a)
Expand ${{\left( b-\frac{x}{2} \right)}^{n}}$ in ascending powers of $x$, where $b$ is a positive constant and $n$ is a negative constant, up to and including the term in ${{x}^{2}}$.
[2]
It is given that the coefficient of $x$ is four times the coefficient of ${{x}^{2}}$ and the constant term in the expansion is $\frac{1}{2}$. Find the integer values of $b$ and $n$.
[3]
(b)
(i) Explain why it is not possible to obtain a Maclaurin series for $\ln \left( 2{{x}^{2}} \right)$.
[1]
(ii) A Taylor series is an expansion of a real function $\text{f}\left( x \right)$ about a point $x=a$ and it is defined by
$\text{f}\left( x \right)=\text{f}\left( a \right)+\text{f}’\left( a \right)\left( x-a \right)+\frac{\text{f}”\left( a \right)}{2!}{{\left( x-a \right)}^{2}}+…+\frac{{{\text{f}}^{\left( n \right)}}\left( a \right)}{n!}{{\left( x-a \right)}^{n}}+…$
where ${{\text{f}}^{\left( n \right)}}\left( a \right)$ is the value of the $n$th derivative of $\text{f}\left( x \right)$ when $x=a$.
Find the first three exact non-zero terms of the Taylor series for $\ln \left( 2{{x}^{2}} \right)$ about the point $x=2$.
[You need not simplify your answer.]
[3]
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