Express $\mathrm{f}(x)=\frac{x-5}{(x+1)(3x+2)}$ in partial fractions and hence obtain \(\mathrm{f}

Question

Express $\mathrm{f}(x)=\frac{x-5}{(x+1)(3x+2)}$ in partial fractions and hence obtain $\mathrm{f}(x)$ as a series in ascending powers of $x$ as far as the term ${{x}^{3}}$. State the range of values of $x$ for which the expansion is valid. Find the coefficient of ${{x}^{n}}$ in the expansion, where $n>1$.

Accepted Answer

$\mathrm{f}(x)=\frac{6}{x+1}-\frac{17}{3x+2}$, $-\frac{5}{2}+\frac{27}{4}x-\frac{105}{8}{{x}^{2}}+\frac{363}{16}{{x}^{3}}+...$, $-\frac{2}{3}<x<\frac{2}{3}$, ${{(-1)}^{n}}\left[ 6-\frac{17}{2}{{\left( \frac{3}{2} \right)}^{n}} \right]$

Maclaurin Series and Power Series - Maclaurin Series and Power Series - General

Maclaurin Series and Power Series - Maclaurin Series and Power Series - General