Find the coefficient of $\frac{1}{{{x}^{r}}}$ in the expansion in descending powers of $x$, \(\f

Question

Find the coefficient of $\frac{1}{{{x}^{r}}}$ in the expansion in descending powers of $x$, $\frac{1}{{{(1+3x)}^{2}}}$ $\frac{1}{{{(1+2x)}^{3}}}$

Accepted Answer

(i) $\displaystyle [x^{-r}](1+3x)^{-2}=(r-1)\left(-\frac13\right)^r$ for integers $r\ge2$, with $|x|>\frac13$. (ii) $\displaystyle [x^{-r}](1+2x)^{-3}=\frac{(-1)^{r+1}(r-1)(r-2)}{2^{r+1}}$ for integers $r\ge3$, with $|x|>\frac12$. In (i) the coefficient is $0$ for integers $r<2$; in (ii) it is $0$ for integers $r<3$.

Maclaurin Series and Power Series - General Terms and Coefficients in Power Series

Maclaurin Series and Power Series - General Terms and Coefficients in Power Series

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