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

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

(a)(i) $\frac{5{{x}^{2}}}{2}-\frac{11}{4}{{x}^{2}}+...$ (ii) $y=\frac{5x}{2}$ (b) $	ext{e}\left( 1+\frac{1}{2}x+\frac{1}{48}{{x}^{3}}+... ight)$

2015 RI Promo Q8

2015 RI Promo Q8

Video Solution:

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