Rewrite $\sum _ { i = 1 } ^ { n } \frac { 9 i ^ { 3 } } { n ^ { 5 } }$

Multiple Choice

Rewrite $\sum _ { i = 1 } ^ { n } \frac { 9 i ^ { 3 } } { n ^ { 5 } }$ as a rational function $S ( n )$ and find $\lim _ { n \rightarrow \infty } S ( n )$ .

A) $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 2 } } , \lim _ { n \rightarrow \infty } S ( n ) = \frac { 9 } { 4 }$
B) $S ( n ) = \frac { 9 n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 } , \lim _ { n \rightarrow \infty } S ( n ) = 9$
C) $S ( n ) = \frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 36 } , \lim _ { n \rightarrow \infty } S ( n ) = 0$
D) $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } } , \lim _ { n \rightarrow \infty } S ( n ) = 0$
E) $S ( n ) = \frac { 9 ( n + 1 ) ^ { 2 } } { 4 n ^ { 3 } }$ , the limit does not exist