Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 11 - Section 11.2 - Series - 11.2 Exercises - Page 717: 72

Answer

a) $S_{n} = D(e^{-at} +e^{-2at} +...+e^{-nat})$ b) $S_{n} = \frac{D}{e^{at}-1}$ c) $D \ge C(e^{at} -1)$

Work Step by Step

a) Let $S_{n}$ denote the residual concentration just before $(n+1)$-st injection. $S_{1} = De^{-at}$ $S_{2} = De^{-at} + De^{-2at}$ $S_{3} = De^{-at} + De^{-2at} + De^{-3at}$ In general $S_{n} = D(e^{-at} +e^{-2at} +...+e^{-nat})$ b) $S_{n} = D(e^{-at} +e^{-2at} +...+e^{-nat})$ $a=De^{-at}$ and $r=e^{-at}$ $nā†’\infty$ $S_{n}=\frac{De^{-at}}{1-e^{-at}}=\frac{D}{e^{at}-1}$ c) $\frac{D}{e^{at}-1} \ge C$ $D \ge C(e^{at} -1)$
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