NYJC/2021/JC2/Prelim/P1/Q07

It is given that $y=\ln \left(1+ e ^x\right)$.
(i) Show that $\left(1+ e ^x\right) \frac{ d y}{ d x}- e ^x=0$

[1]

(ii) By further differentiation of the result in (i), find the Maclaurin series for $y$, up to and including the term in $x^2$. Hence find the series for $\frac{ e ^x}{1+ e ^x}$ up to and including $x$.

[4]

(iii) Using appropriate expansion from the List of Formulae (MF26), verify the correctness of the series $\frac{ e ^x}{1+ e ^x}$ found in (ii).

[3]