Find the coefficient of ${{x}^{r}}$ in the expansion in ascending powers of $x$, \(\frac{1}{2-{{

Question

Find the coefficient of ${{x}^{r}}$ in the expansion in ascending powers of $x$, $\frac{1}{2-{{x}^{2}}}$ $\frac{2}{2-x}$

Accepted Answer

(i) $\displaystyle [x^r]\frac{1}{2-x^2}=\begin{cases}2^{-(r/2+1)},&r\text{ even},\0,&r\text{ odd},\end{cases}$ for integers $r\ge0$. Equivalently, the coefficient of $x^{2k}$ is $2^{-(k+1)}$. (ii) $\displaystyle [x^r]\frac{2}{2-x}=\frac{1}{2^r}$ for integers $r\ge0$.

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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