Chapter 5 - Section 5.2 - Sigma Notation and Limits of Finite Sums - Exercises - Page 300: 43

$\dfrac{5}{6}$

Here, we have $f(x)=x+x^2$ Then, $\Sigma_{i=1}^n (\dfrac{1}{n}) (c_i+c_i^2)=(\dfrac{1}{n}) \Sigma_{i=1}^n (\dfrac{i}{n}+\dfrac{i^2}{n^2})$ or, $(\dfrac{1}{n^3})\Sigma_{i=1}^n i+(\dfrac{1}{n^3})\Sigma_{i=1}^n i^2=\dfrac{5n^3+6n^2+n}{6n^3}$ Thus, $\Sigma_{i=1}^n (\dfrac{1}{n}) (c_i+c_i^2)=\lim\limits_{n \to \infty}\dfrac{5+6/n+1/n^2}{6}=\dfrac{5}{6}$