Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 11 - Infinite Sequences and Series - 11.1 Exercises - Page 724: 22

Answer

The sequence diverges

Work Step by Step

We are given the sequence: $a_n=1+\dfrac{10^n}{9^n}$. Determine the first 10 terms: $a_1=1+\dfrac{10^1}{9^1}\approx 2.11111$ $a_2=1+\dfrac{10^2}{9^2}\approx 2.23457$ $a_3=1+\dfrac{10^3}{9^3}\approx 2.37174$ $a_4=1+\dfrac{10^4}{9^4}\approx 2.52416$ $a_5=1+\dfrac{10^5}{9^5}\approx 2.69351$ $a_6=1+\dfrac{10^6}{9^6}\approx 2.88168$ $a_7=1+\dfrac{10^7}{9^7}\approx 3.09075$ $a_8=1+\dfrac{10^8}{9^8}\approx 3.32306$ $a_9=1+\dfrac{10^9}{9^9}\approx 3.58117$ $a_{10}=1+\dfrac{10^{10}}{9^{10}}=3.86797$ The sequence doesn't appear to have a limit. $a_n=1+\dfrac{10^n}{9^n}=1+\left(\dfrac{10}{9}\right)^n$ As $\left(\dfrac{10}{9}\right)^n$ is increasing when $n$ is increasing, $a_n$ also increases, so we have: $\displaystyle{\lim_{n \to \infty}} \left[1+\left(\dfrac{10}{9}\right)^n\right]=1+\displaystyle{\lim_{n \to \infty}}\left(\dfrac{10}{9}\right)^n$ $=1+\infty=\infty$ Plot the first 10 terms of the sequence:
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