Question

Sum $1+3x+5x^2+7x^3+9x^4+....$ to infinity.

Answer 1

Let $S=1+3x+5x^2+7x^3......\infty$
Multiplying both sides by $a$, we get
$Sx=0+x+3x^2+5x^3.......\infty$
Subtract both eqns, we get
$S(1-x)=1+2(x+x^2+x^3.......\infty )$
Clearly, the series in the bracket is a G.P. with common ratio $=x$
$\therefore S(1-x)=1+\frac{2x}{1-x}=\frac{1+x}{1-x}$
$\Rightarrow S=\frac{1+x}{(1-x{)}^2}$