Question

Find the sum of the infinite series $\frac{1.2}{3}+\frac{2.3}{3^2}+\frac{3.4}{3^3}+\frac{4.5}{3^4}+....$

Answer 1

General term of the series in $n(n+1)x^n$, where $x=\frac{1}{3}$

Therefore, $S=2x+6x^2+12x^3+20x^4+\dots \dots$

$\Rightarrow -3xS=-6x^2-18x^3-36x^4-\dots$

$\Rightarrow 3x^{2}S=6x^3+18x^4+\dots$

$\Rightarrow -x^{3}S=-2x^4-\dots$

Adding the above series, we get

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

$\Rightarrow (1-x{)}^{3}S=2x$

$\Rightarrow S=\frac{2x}{(1-x{)}^3}=\frac{9}{4}[\because x=\frac{1}{3}]$