lim(n??)??_(i=1)^n?(1+2i/n)^3 (2/n) ?
=2/n ?_(i=1)^n?(1+2i/n)(1+2i/n)(1+2i/n) ?
2/n ?_(i=1)^n?(1+4i/n+(2i^2)/n^2 )(1+2i/n) ?
2/n ?_(i=1)^n?(1+2i/n+4i/n+(2i^2)/n^2 +(8i^2)/n^2 +(4i^3)/n^3 ) ?
2/n ?_(i=1)^n?(1+6i/2n+(10i^2)/(2n^2 )+(4i^3)/n^3 ) ?
2/n (?_(i=1)^n?1+?_(i=1)^n 6i/2n+?_(i=1)^n (10i^2)/(2n^2 )+?_(i=1)^n (4i^3)/n^3 ?)
2/n (?_(i=1)^n?1+3/n ?_(i=1)^n i+5/n^2 ?_(i=1)^n i^2 +4/n^3 ?_(i=1)^n i^3 ?)
2/n (4/n^3 [(n^2 (n+1)^2)/4]+5/n^2 [n(n+1)(2n+1)/6]+3/n [n(n+1)/2]+1n)
8/4 [(n^2 (n+1)^2)/n^4 ]+10/6 [n(n+1)(2n+1)/n^3 ]+6/2 [n(n+1)/n^2 ]+2
24/12+20/12+36/12+24/12=104/12=52/6
All better now?
=2/n ?_(i=1)^n?(1+2i/n)(1+2i/n)(1+2i/n) ?
2/n ?_(i=1)^n?(1+4i/n+(2i^2)/n^2 )(1+2i/n) ?
2/n ?_(i=1)^n?(1+2i/n+4i/n+(2i^2)/n^2 +(8i^2)/n^2 +(4i^3)/n^3 ) ?
2/n ?_(i=1)^n?(1+6i/2n+(10i^2)/(2n^2 )+(4i^3)/n^3 ) ?
2/n (?_(i=1)^n?1+?_(i=1)^n 6i/2n+?_(i=1)^n (10i^2)/(2n^2 )+?_(i=1)^n (4i^3)/n^3 ?)
2/n (?_(i=1)^n?1+3/n ?_(i=1)^n i+5/n^2 ?_(i=1)^n i^2 +4/n^3 ?_(i=1)^n i^3 ?)
2/n (4/n^3 [(n^2 (n+1)^2)/4]+5/n^2 [n(n+1)(2n+1)/6]+3/n [n(n+1)/2]+1n)
8/4 [(n^2 (n+1)^2)/n^4 ]+10/6 [n(n+1)(2n+1)/n^3 ]+6/2 [n(n+1)/n^2 ]+2
24/12+20/12+36/12+24/12=104/12=52/6
All better now?