Rewrite $\sum _ { i = 1 } ^ { n } \left[ \frac { 3 } { n } + \left( \frac { 4 i } { n ^ { 2 } } \right) \right] \left( \frac { 4 i } { n } \right)$

Multiple Choice

Rewrite $\sum _ { i = 1 } ^ { n } \left[ \frac { 3 } { n } + \left( \frac { 4 i } { n ^ { 2 } } \right) \right] \left( \frac { 4 i } { n } \right)$ as a rational function S(n) and find $\lim _ { n \rightarrow \infty } S ( n )$ .

A) $S ( n ) = \frac { 2 [ 9 n ( n + 1 ) + 4 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } } \text {, limit dos not exist }$
B) $S ( n ) = \frac { 2 [ 9 + 4 n ] } { 3 n } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 34 } { 3 }$
C) $S ( n ) = \frac { 2 [ 9 n + 4 ( n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 34 } { 3 }$
D) $S ( n ) = \frac { 2 [ 9 n ( n + 1 ) + 4 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 5 } { 6 }$
E) $S ( n ) = \frac { 2 [ 9 n ( n + 1 ) + 4 ( n + 1 ) ( 2 n + 1 ) ] } { 3 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 34 } { 3 }$