If |x|<1, then the sum of the series
1 + 2x + 3x2 + 4x3 + ........., ∞ will be
This is an A.G.P.
Let S = 1 + 2x + 3x2 + ........ ∞
⇒ x.S = x + 2x2 + .......... ∞
Subtracting (1 - x)S = 1 + x + x2 + ............. ∞ = 11−x
∴ S = 1(1−x)2.
Aliter: Use S = [1 + r1−r × diff. of A.P.]11−r.