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Many are ensuring that ¼ is the value of the limit. That is not in the option however.

Many are ensuring that ¼ is the value of the limit. That is not in the option however.

Question Image
Grade:12th pass

1 Answers

Aditya Gupta
2075 Points
one year ago
simply write the expression inside limit as
(∑r=1n r*∑k=1n-r+1 k)/n^4
= (∑r=1n r*(n – r+1)(n – r+2))/2n^4
apply KINGS rule to simplify this as
(∑r=1n (1+n – r)*(r)(r+1))/2n^4
= (∑r=1n r(r+1)(n+1 – r))/2n^4
now simply open the brackets r(r+1)(n+1 – r) and find its sum (since ∑r, ∑r^2, ∑r^3 are known) and then finally apply the limit.
you will get the ans as (1/3 – 1/4)/2
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