RI Maclaurin Series Tutorial Q4

Given that \(y=\tan \left( {{\mathrm{e}}^{2x}}-1 \right)\), show that \(\frac{\mathrm{d}y}{\mathrm{d}x}=k{{\mathrm{e}}^{2x}}\left( 1+{{y}^{2}} \right)\), where \(k\) is to be found. Hence find the values of \(\frac{\mathrm{d}y}{\mathrm{d}x}\), \(\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}\) and \(\frac{{{\mathrm{d}}^{3}}y}{\mathrm{d}{{x}^{3}}}\) when \(x=0\).[6]

Write down the first three non-zero terms in the Maclaurin series for \(\tan \left( {{\mathrm{e}}^{2x}}-1 \right)\).[1]

The first two non-zero terms in the Maclaurin series for \(\tan \left( {{\mathrm{e}}^{2x}}-1 \right)\) are equal to the first two non-zero terms in the series expansion of \({{\mathrm{e}}^{ax}}\ln \left( 1+nx \right)\). By using appropriate expansions from the List of Formulae (MF26), find the constants \(a\) and \(n\). Hence find the third non-zero term of the series expansion of \({{\mathrm{e}}^{ax}}\ln \left( 1+nx \right)\) for these values of \(a\) and \(n\).[4]