Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.2 Series - 11.2 Exercises: 4


sum of $\Sigma a_{n}$ (where n is an integer starting at 1 and approaching infinity) is: $\lim\limits_{n\to \infty} s_{n}$ = $\lim\limits_{n\to \infty} \frac{(n^2) -1 }{4(n^4) + 1} $ = $\lim\limits_{n\to \infty} \frac{n^2} {4(n^2) } \frac{1 -\frac{1 }{n^2} } {1 + \frac{1 }{n^2} } $= $\frac{1}{4}$

Work Step by Step

Recall the Definition 2 from page 748 in our textbook, taking the limit we will get: $\lim\limits_{n\to \infty} s_{n}$ = $\lim\limits_{n\to \infty} \frac{(n^2) -1 }{4(n^2) + 1} $ = $\frac{1}{4}$. Now, how we arrived at $\frac{1}{4}$as an answer only requires that we take the highest order of terms as common factor from denominator and numerator and then take the limit. We get $ \frac{n^2} {4(n^2) } \frac{1 -\frac{1 }{n^2} } {1 + \frac{1 }{n^2} }$ ; then $\frac{n^2} {4(n^2) }$ equals $ \frac{1} {4 }$ , taking the limit of this fraction for n approaching infinity gives us $\frac{1}{4}$and thus the whole summation is $\frac{1}{4}$.
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