Let Pn = {(2^3 – 1)/(2^3 + 1)}*{(3^3 – 1)/(3^3 + 1)}*…..*{(n^3 – 1)/(n^3 + 1)}; n = 2, 3, ….. lim Pn as n - > ∞ is

(a) ¾

(b) 7/11

(c) 2/3

(d) ½

Ans is c. but i dont know how to approach them.

Let’s factorize the numerator and denominators:

Numerator will be:

(2–1)(2^2+2+1)(3–1)(3^2+3+1)…(n-1)(n^2+n+1) = (1*7)(2*13)(3*21)…((n-1)*(n^2+n+1))

Similarly, denominator is:

(2+1)(2^2-2+1)(3+1)(3^2-3+1)…(n+1)(n^2-n+1) = (3*3)(4*7)(5*13)…((n+1)*(n^2-n+1))

Let’s look at the first element inside every parenthesis in numerator and denominator. Most of them will cancel out. Second element inside parenthesis also will be cancelling out as (k^2+k+1) = (k+1)^2 - (k+1) + 1.

So we are left with following fraction:

1*2*(n^2+n+1)/n*(n+1)*3 = (2n^2+2n+2)/(3n^2+3n)

When n tends to infinity, this will be 2/3