Expand $\mathrm{f}\left( x \right)=\frac{2}{2-x}-\frac{1}{{{\left( 1+x \right)}^{2}}}$ as a series i

Question

Expand $\mathrm{f}\left( x \right)=\frac{2}{2-x}-\frac{1}{{{\left( 1+x \right)}^{2}}}$ as a series in ascending powers of $x$ up to and including the term in ${{x}^{2}}$. [3] State the equation of the tangent to the curve $y=\mathrm{f}\left( x \right)$ at the origin. [1] Using the standard series given in the List of Formulae (MF15) or otherwise, show that the first three non- zero terms in the Maclaurin series for ${{\mathrm{e}}^{\sqrt{1+x}}}$ can be expressed as $\mathrm{e}\left( 1+px+q{{x}^{3}}+... \right)$ where $p$ and $q$ are constants to be determined. [6]

Accepted Answer

$f(x) = \frac{5}{2}x - \frac{11}{4}x^2 + \ldots$; $y = \frac{5}{2}x$; $p = \frac{1}{2}, \; q = \frac{1}{48}$

2015 RI Promo Q8

2015 RI Promo Q8

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