_ n → ∞ 1 2 ^ n ( 1 1 - 1 2 ^ n + 1 1 - 2 2 ^ n + 1 1 - 3 2 ^ n + ....

Let $I = \lim_{n \to \infty}\dfrac{1}{{2}^{n}}\left(\dfrac{1}{\sqrt{1 - \dfrac{1}{{2}^{n}}}} + \dfrac{1}{\sqrt{1 - \dfrac{2}{{2}^{n}}}} + \dfrac{1}{\sqrt{1 - \dfrac{3}{{2}^{n}}}} + \ldots . + \dfrac{1}{\sqrt{1 - \dfrac{{2}^{n} - 1}{{2}^{n}}}}\right)$

Let ${2}^{n} = t$ and if $n \to \infty$ then $t \to \infty$

$I = \lim_{n \to \infty}\dfrac{1}{t}\left(\mathop{∑}_{r = 1}^{t - 1}\dfrac{1}{\sqrt{1 - \dfrac{r}{t}}}\right)$

$I = {∫}_{0}^{1}\dfrac{dx}{\sqrt{1 - x}}$ (Using definite integral as limit of sum)

Applying ${∫}_{0}^{a}f\left(x\right)dx = {∫}_{0}^{a}f\left(a - x\right)dx$, we get

$I = {∫}_{0}^{1}\dfrac{dx}{\sqrt{x}}$

$= {\left[2\sqrt{x}\right]}_{0}^{1} = 2$