<ol class="abc-list"><li><div class="list-item-paragraph">Given that&nbsp;\(y = \sqrt{\ln(x + \mathrm{e})}\) for&nbsp;\(x &gt; 1 - \mathrm{e}\), show that</div><div class="list-item-paragraph" style="text-align: center">\[2y\frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{e}^{-y^2}.\]</div><div class="list-item-paragraph">Hence find the first three terms of the Maclaurin expansion of \(y\). Give the coefficients as exact fractions in their simplest form.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Using standard series from the List of Formulae (MF27), expand \(\ln(x + \mathrm{e})\) as far as the term in \(x^2\).</div><div class="list-item-paragraph">Hence, verify the correctness of your expansion in <strong>(a)</strong>, showing your working clearly.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Use your expansion in part <strong>(a)</strong> to find an approximation to \(\sqrt{\ln\left(\frac{1 + 10\mathrm{e}}{10}\right)}\), leaving your answer in terms of \(\mathrm{e}\).<span class="inline-marks">[2]</span></div></li></ol>

Given that \(y = \sqrt{\ln(x + \mathrm{e})}\) for \(x > 1 - \mathrm{e}\), show that

\[2y\frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{e}^{-y^2}.\]

Hence find the first three terms of the Maclaurin expansion of \(y\). Give the coefficients as exact fractions in their simplest form.

Using standard series from the List of Formulae (MF27), expand \(\ln(x + \mathrm{e})\) as far as the term in \(x^2\).

Hence, verify the correctness of your expansion in **(a)**, showing your working clearly.

Use your expansion in part **(a)** to find an approximation to \(\sqrt{\ln\left(\frac{1 + 10\mathrm{e}}{10}\right)}\), leaving your answer in terms of \(\mathrm{e}\).

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