2021 NYJC P1 Q7
It is given that $y=\ln \left( 1+{{\text{e}}^{x}} \right)$.
(i)
Show that $\left( 1+{{\text{e}}^{x}} \right)\frac{\text{d}y}{\text{d}x}-{{\text{e}}^{x}}=0$.
[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 $\frac{{{\text{e}}^{x}}}{1+{{\text{e}}^{x}}}$ up to and including $x$.
[4]
(iii)
Using appropriate expansion from the List of Formulae (MF26), verify the correctness of the series $\frac{{{\text{e}}^{x}}}{1+{{\text{e}}^{x}}}$ found in (ii).
[3]
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