Expand $\frac{{{x}^{2}}+2x}{2{{x}^{2}}+1}$ in ascending powers of $x$ up to and including the term i

Question

Expand $\frac{{{x}^{2}}+2x}{2{{x}^{2}}+1}$ in ascending powers of $x$ up to and including the term in ${{x}^{5}}$. State the range of values of $x$ for which this expansion is valid. [3] Find, in the simplest form, the coefficient of ${{x}^{2017}}$ in this expansion. [2]

Accepted Answer

$2x+x^2-4x^3-2x^4+8x^5+\cdots$, valid for $-\frac1{\sqrt2}<x<\frac1{\sqrt2}$; coefficient of $x^{2017}$ is $2^{1009}$

2009 HCI Promo P1 Q3 (modified)

2009 HCI Promo P1 Q3 (modified)

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