It is given that $y = \mathrm{e}^{\tan x}$, for \(-\frac{\pi}{2}

Question

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 $\frac{y_1}{y} = 1 + \tan^2 x$.[2] By expressing $\ln y_1$ in terms of $\tan x$, show that $\frac{y_2}{y_1} = (1 + \tan x)^2$.[3] Find $\frac{y_3}{y_2}$ in terms of $\tan x$.[3] Hence obtain the Maclaurin series for $\mathrm{e}^{\tan x}$ in ascending powers of $x$ up to and including the term in $x^3$.[2]

Accepted Answer

(a) $\dfrac{y_1}{y}=1+\tan^2x$ (b) $\dfrac{y_2}{y_1}=(1+\tan x)^2$ (c) $\dfrac{y_3}{y_2}=(1+\tan x)^2+\dfrac{2(1+\tan^2x)}{1+\tan x}$ (d) $\mathrm e^{\tan x}=1+x+\dfrac12x^2+\dfrac12x^3+\cdots$

Maclaurin Series and Power Series - Maclaurin Series from Differential Equations

Maclaurin Series and Power Series - Maclaurin Series from Differential Equations

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