University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

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



Work Step by Step

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, $\Sigma_{i=1}^n (\dfrac{1}{n}) (c_i+c_i^2)=\lim\limits_{n \to \infty}\dfrac{5+6/n+1/n^2}{6}=\dfrac{5}{6}$
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