It is given that $y=\sqrt{1+\ln (1+x)}.$ By differentiating successively, find the Maclaurin’

Question

It is given that $y=\sqrt{1+\ln (1+x)}.$ By differentiating successively, find the Maclaurin’s series for $y,$ up to and including the term in ${{x}^{2}}.$[3] By substituting $x=\frac{1}{4},$approximate $\ln \left( \frac{4}{5} \right)$, leaving your answer in exact form.[3]

Accepted Answer

$\text{(i)}\ 1+\frac12x-\frac38x^2+O(x^3)$ $\text{(ii)}\ \displaystyle \ln\frac45\approx-\frac{3497}{16384}$

VJC P1 Q2

VJC P1 Q2

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