Express $\mathrm{f}(x)=\frac{16-9x}{(1-3x)(4+x)}$ in partial fractions and hence obtain \(\mathrm{

Question

Express $\mathrm{f}(x)=\frac{16-9x}{(1-3x)(4+x)}$ in partial fractions and hence obtain $\mathrm{f}(x)$ as a series in ascending powers of $x$, giving the first $3$ non-zero terms of this expansion. Find the coefficient of${{x}^{r}}$ in the expansion.

Accepted Answer

$f(x)=\dfrac3{1-3x}+\dfrac4{4+x}=4+\dfrac{35}{4}x+\dfrac{433}{16}x^2+\cdots$ for $|x|<\dfrac13$; coefficient of $x^r$ is $3^{r+1}+\left(-\dfrac14\right)^r$.

Maclaurin Series and Power Series - Binomial Expansion as a Maclaurin Series

Maclaurin Series and Power Series - Binomial Expansion as a Maclaurin Series

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