It is given that $\mathrm{f}\left( x \right)=\ln \left( 1+{{\mathrm{e}}^{-x}} \right)$. Using standa

Question

It is given that $\mathrm{f}\left( x \right)=\ln \left( 1+{{\mathrm{e}}^{-x}} \right)$. Using standard series from the List of Formulae (MF26) or otherwise, show that the first three non- zero terms in the Maclaurin series for $\mathrm{f}\left( x \right)$ is $\ln a+bx+c{{x}^{2}}$, where $a$, $b$ and $c$ are constants to be determined. [4] Hence find an expansion for $\frac{1}{1+{{\mathrm{e}}^{x}}}$, up to and including the term in $x$. [2]

Accepted Answer

(i) $a = 2$, $b = -\frac{1}{2}$, $c = \frac{1}{8}$; (ii) $\frac{1}{1+e^x} \approx \frac{1}{2} - \frac{1}{4}x$

2023 JPJC P2 Q1

2023 JPJC P2 Q1

Video Solution:

Solution:

Solution locked