Answer
$S= 1.2 , 1.02, 1.002, 1.0002$
When $n=10, 100, 1000, 10000$
Work Step by Step
$\Sigma\frac{2i+1}{n^2}$
$\frac{1}{n^2}\Sigma{2i+1}$, take out the $\frac{1}{n^2}$
$\frac{1}{n^2}\Sigma{2i} + \Sigma1 $, separate into two summations
$\frac{1}{n^2}[(2\times\frac{n(n+1)}{2}) + n]$, use summation formulas
$\frac{1}{n^2}\times(n^2+2n)$, simplify
$(1+\frac{2}{n})$, distribute
Substitute the n values
When $n=10, S=(1+\frac{2}{10})= 1.2$
When $n=100, S=(1+\frac{2}{100})= 1.02$
When $n=1000, S=(1+\frac{2}{1000})= 1.002$
When $n=10000, S=(1+\frac{2}{10000})= 1.0002$
