Differentation

$y=1+x+x^2+x^3+\ldots$

Since $|x|<1$

$y=\dfrac{1}{1-x}$

$\begin{aligned} \dfrac{dy}{dx}&=\dfrac{d}{dx}[(1-x)^{-1}]\\ &=-1(1-x)^{-2}\cdot\dfrac{d}{dx}[1-x]\\ &=(1-x)^{-2} \end{aligned}$

$\Longrightarrow k=\boxed{2}$