Progression and series

Question

1*2+2*3x+3*4x^2+........up to infinity .find

Its sum if |x|<1.

Comments

$\frac{2}{1-x}$

Answer 1

S = 1*2*x⁰ + 2*3*x¹+3*4*x² + ......+ infinity

take integration of both side:

Sx = 1*2* $\frac{x1}{1}$ + 2*3* $\frac{x2}{2}$+3*4* $\frac{x3}{3}$ + ......+ infinity

take again integration both side:

S$\frac{x2}{2}$= x²+x³+x⁴+x⁵+.....+infinity

$\frac{S}{2}$=1+x+x²+x³+....+infinity

S=$\frac{2}{1-x}$