l i m n → ∞ ( n + 1 ) 1 / 3 n 4 / 3 + ( n + 2 ) 1 / 3 n 4 / 3 + . . . . . + ( 2 n ) 1 / 3 n 4 /…

limn→∞(n+1)1/3n4/3+(n+2)1/3n4/3+.....+(2n)1/3n4/3 is equal to
  1. 34(2)4/3-34
  2. 43(2)3/4
  3. 43(2)4/3
  4. 34(2)4/3-43

Solution

The given limit can be written as
limn→∞1n1+1n13+1+2n13+….+1+nn13
=limn→∞1n∑r=1n1+rn13
=∫01(1+x)1/3dx
=1+x434301
=34(24/3–1)
=34.24/3-34

Asked in: JEE Main 2019 (10 Apr Shift 1)

Practice more Definite Integration questions on Aicharya