Question

If |x|<1, then the sum of the series

1 + 2x + 3$x^2$ + 4$x^3$ + ........., ∞ will be

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

This is an A.G.P.

Let S = 1 + 2x + 3$x^2$ + ........ ∞

⇒ x.S = x + 2$x^2$ + .......... ∞

Subtracting (1 - x)S = 1 + x + $x^2$ + ............. ∞ = $\frac{1}{1-x}$

∴ S = $\frac{1}{(1-x{)}^2}$.

Aliter: Use S = [1 + $\frac{r}{1-r}$ × diff. of A.P.]$\frac{1}{1-r}$.