l i m n → ∞ ( n + 1 ) 1 / 3 n 4 / 3 + ( n + 2 ) 1 / 3 n 4 / 3 + . . . . . + ( 2 n ) 1 / 3 n 4 / 3 is equal to

Question

l i m n → ∞ ( n + 1 ) 1 / 3 n 4 / 3 + ( n + 2 ) 1 / 3 n 4 / 3 + . . . . . + ( 2 n ) 1 / 3 n 4 / 3 is equal to, 3 4 ( 2 ) 4 / 3 - 3 4, 4 3 ( 2 ) 3 / 4, 4 3 ( 2 ) 4 / 3, 3 4 ( 2 ) 4 / 3 - 4 3

MARKS App · Accepted Answer

The given limit can be written as l i m n → ∞ 1 n 1 + 1 n 1 3 + 1 + 2 n 1 3 + … . + 1 + n n 1 3 = l i m n → ∞ 1 n ∑ r = 1 n 1 + r n 1 3 = ∫ 0 1 ( 1 + x ) 1 / 3 d x = 1 + x 4 3 4 3 0 1 = 3 4 ( 2 4 / 3 – 1 ) = 3 4 . 2 4 / 3 - 3 4

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