Expand $\frac{5x+6}{(2+x)(1-x)}$ in ascending powers of $x$ and write coefficient of $x^n$. Also state the condition under which the expansion is valid.
I got coefficient of $x^n$ as $(-1)^n.(2/3).2^n + 11/3$. expression is valid if mod(x)<1 and mod(x/2)<1. is my solution correct? Kindly help.
Write $$\frac{5x+6}{(2+x)(1-x)} = \frac{\frac52 x + 3}{\left(1+\frac12 x\right)(1-x)}. $$ By partial fraction decomposition, this is equal to $$-\frac23\left(\frac1{1+\frac12 x}\right) + \frac{11}3\left(\frac1{1-x}\right). $$ The series expansion is $$\sum_{n=0}^\infty \left(\frac{11}3 - \frac23\left(-\frac12\right)^n \right)x^n. $$ Hence the coefficient of $x^n$ is $\frac{11}3 - \frac23\left(-\frac12\right)^n$, and this is valid when $|x|<1$.