<p>It is given that&nbsp;\(y = \mathrm{e}^{\tan x}\), for&nbsp;\(-\frac{\pi}{2} &lt; x &lt; \frac{\pi}{2}\), and \(y_r = \frac{\mathrm{d}^r y}{\mathrm{d}x^r}\), for&nbsp;\(r = 1, 2, 3\).</p>
<ol class="abc-list"><li data-list-item-id="part-a"><div class="list-item-paragraph">Show that&nbsp;\(\frac{y_1}{y} = 1 + \tan^2 x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="part-b"><div class="list-item-paragraph">By expressing \(\ln y_1\) in terms of \(\tan x\), show that&nbsp;\(\frac{y_2}{y_1} = (1 + \tan x)^2\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="part-c"><div class="list-item-paragraph">Find \(\frac{y_3}{y_2}\) in terms of \(\tan x\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="part-d"><div class="list-item-paragraph">Hence obtain the Maclaurin series for&nbsp;\(\mathrm{e}^{\tan x}\) in ascending powers of \(x\) up to and including the term in \(x^3\).<span class="inline-marks">[2]</span></div></li></ol>

It is given that \(y = \mathrm{e}^{\tan x}\), for \(-\frac{\pi}{2} < x < \frac{\pi}{2}\), and \(y_r = \frac{\mathrm{d}^r y}{\mathrm{d}x^r}\), for \(r = 1, 2, 3\).

Show that \(\frac{y_1}{y} = 1 + \tan^2 x\).

By expressing \(\ln y_1\) in terms of \(\tan x\), show that \(\frac{y_2}{y_1} = (1 + \tan x)^2\).

Find \(\frac{y_3}{y_2}\) in terms of \(\tan x\).

Hence obtain the Maclaurin series for \(\mathrm{e}^{\tan x}\) in ascending powers of \(x\) up to and including the term in \(x^3\).

Maclaurin Series and Power Series

Maclaurin Series and Power Series