In this problem a and r will represent constants.(a) Briefly explain (just with words) why lim r" = 0 if Ir|<1.%3Dn-00(b) Expand the following: (1 – r)(a + ar + ar + ar' +.. + ar")(c) For Ir|<1, find the limit as n → ∞ of the previous part and explain how this gives the formula for a sum of a00geometric series: > ar"%3D1-rn=0

For the first question we use r<1 implies (1/r)>1.

In this problem a and r will represent constants. (a) Briefly explain (just with words) why lim r" = 0 if |r|<1. %3D n-00 (b) Expand the following: (1 – r)(a + ar + ar + ar' + .. + ar") (c) For r|<1, find the limit as n o of the previous part and explain how this gives the formula 00 geometric series: > ar" 1-r n=0

In this problem a and r will represent constants. (a) Briefly explain (just with words) why lim r" = 0 if |r|<1. %3D n-00 (b) Expand the following: (1 – r)(a + ar + ar + ar' + .. + ar") (c) For r|<1, find the limit as n o of the previous part and explain how this gives the formula 00 geometric series: > ar" 1-r n=0

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

In this problem a and r will represent constants.
(a) Briefly explain (just with words) why lim r" = 0 if Ir|<1.
%3D
n-00
(b) Expand the following: (1 – r)(a + ar + ar + ar' +.. + ar")
(c) For Ir|<1, find the limit as n → ∞ of the previous part and explain how this gives the formula for a sum of a
00
geometric series: > ar"
%3D
1-r
n=0

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