Question

Sum of the series $0.2+0.004+0.00006+........\infty$

Answer 1

$S=0.2+0.004+0.00006+....\infty$

$S=2\times {10}^{-1}+4\times {10}^{-3}+6\times {10}^{-5}+8\times {10}^{-7}+....$

Let ${10}^{-1}=x$

$S=\left(\frac{1-x^2}{1+x^2}\right)(2x+4x^3+6x^5+8x^7+.....)$

$=\left(\frac{1}{1-x^2}\right)(2x-2x^3+4x^3-4x^5+6x^5-6x^7+....)$

$S(1-x^2)=2x+2x^3+2x^5+2x^2+....$

$2(x+x^3+x^5+......)$

$S(1-x^2)=\frac{2x}{1-x^2}$

$S=\frac{2\times {10}^{-1}}{{(1-{10}^{-2})}^2}$

$=\frac{2000}{5807}$