Question

`lim_(n rightarrow ∞)1/2^n [1/sqrt(1 - 1/2^n) + 1/sqrt(1 - 2/2^n) + 1/sqrt(1 - 3/2^n) + ...... + 1/sqrt(1 - (2^n - 1)/2^n)]` is equal to ______.

Options

• `1/2`

• 1

• 2

• –2

Answer 1

`lim_(n rightarrow ∞)1/2^n [1/sqrt(1 - 1/2^n) + 1/sqrt(1 - 2/2^n) + 1/sqrt(1 - 3/2^n) + ...... + 1/sqrt(1 - (2^n - 1)/2^n)]` is equal to 2.

Explanation:

`lim_(n rightarrow ∞)1/2^n [1/sqrt(1 - 1/2^n) + 1/sqrt(1 - 2/2^n) + 1/sqrt(1 - 3/2^n) + ...... + 1/sqrt(1 - (2^n - 1)/2^n)]`

S = `lim_(n rightarrow ∞) sum_(r = 1)^(2^n - 1) 1/2^n[1/sqrt(1 - r/2^n)]`

Let `r/2^n` = x and `1/2^n` = dx

when r = 1 and n `rightarrow` ∞ then x `rightarrow` 0

When r = 2ⁿ – 1 and n `rightarrow` ∞ then

x = `lim_(n rightarrow ∞) (2^n - 1)/2^n = lim_(n rightarrow ∞) 1 - 1/2^n` = 0

∴ S = `int_0^1 1/sqrt(1 - x)dx`

= `-2[sqrt(1 - x)]_0^1`

= 2