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Question:

lim_{n→∞}((n+1)^(1/3)n^(4/3) + (n+1)^(1/3)n^(4/3) + .... + (2n)^(1/3)n^(4/3)) is equal to:

43(2)4/3

34(2)4/3−34

34(2)4/3−43

43(2)3/4

Solution:

Correct option is C. 34(2)^4/3−43
Given lim_{n→∞}((n+1)^(1/3)n^(4/3)+(n+1)^(1/3)n^(4/3)+…+(2n)^(1/3)n^(4/3))
lim_{n→∞} Σ_{r=1}^n (1/n)(n+rn)^(1/3) = ∫_0^1 (1+x)^(1/3)dx = [(1+x)^(4/3)/4/3]_0^1 = 3/4(2^(4/3)−1);