Explanation

(d)
Let $S = 1 + 2x + 3x^{2} + 4x^{3} + ...\infty$ …(i)
$S = x + 2x^{2} + 3x^{3} + ...\infty$ …(ii)
On subtracting Eq. (ii) from Eq. (i), we get
$(1 - x)S = (1 + x + x^{2} + ...\infty)$
$\Rightarrow S = \frac{1}{(1 - x)}\left( \frac{1}{1 - x} \right) = \frac{1}{(1 - x)^{2}}$
Alternate Here, $a = 1,\ d = 1,\ r = x$
$\therefore S_{\infty} = \frac{a}{1 - r} + \frac{d.r}{(1 - r)^{2}}$
$= \frac{1}{1 - x} + \frac{1.x}{(1 - x)^{2}} = \frac{1}{(1 - x)^{2}}$