displaystyle limn→∞ (((n+1)(1/3) /n(4)3) + ((n+2)(1/3)/n(4)3) + ..... + ((2n)(1/3)/n(4)3)) equal to :

Question

displaystyle limn→∞ (((n+1)(1/3) /n(4)3) + ((n+2)(1/3)/n(4)3) + ..... + ((2n)(1/3)/n(4)3)) equal to : (A) (4/3) (2)(4/3)  (B) (3/4) (2)(4/3) - (4/3)  (C) (3/4) (2)(4/3) - (3/4)  (D) (4/3) (2)(3/4)

Tardigrade Admin · Accepted Answer

limn→∞ ∑nr=1 (1/n) ((n+r/n))1/3 = ∫10 (1+x)1/3 dx = (3/4) (24/3 -1)

Q. $n \to \infty lim (\frac{( n + 1 ) ^{\frac{1}{3}}}{n ^{\frac{4}{3}}} + \frac{( n + 2 ) ^{\frac{1}{3}}}{n ^{\frac{4}{3}}} + ..... + \frac{( 2 n ) ^{\frac{1}{3}}}{n ^{\frac{4}{3}}})$ equal to :

$\frac{4}{3} (2)^{\frac{4}{3}}$

$\frac{3}{4} (2)^{\frac{4}{3}} - \frac{4}{3}$

$\frac{3}{4} (2)^{\frac{4}{3}} - \frac{3}{4}$

$\frac{4}{3} (2)^{\frac{3}{4}}$

$lim_{n \to \infty} \sum_{r = 1}^{n} \frac{1}{n} (\frac{n + r}{n})^{1/3}$
$= \int_{0}^{1} (1 + x)^{1/3} d x = \frac{3}{4} (2^{4/3} - 1)$

Q. n→∞lim​(n34​(n+1)31​​+n34​(n+2)31​​+.....+n34​(2n)31​​) equal to :

34​(2)34​

43​(2)34​−34​

43​(2)34​−43​

34​(2)43​