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SANDS10

# SANDS10 - Sequences and Series An Introduction to...

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Sequences and Series: An Introduction to Mathematical Analysis by Malcolm R. Adams c circlecopyrt 2010

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Contents 1 Sequences 1 1.1 The general concept of a sequence . . . . . . . . . . . . . . . . 1 1.2 The sequence of natural numbers . . . . . . . . . . . . . . . . 12 1.3 Sequences as functions . . . . . . . . . . . . . . . . . . . . . . 22 1.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5 Tools for Computing Limits . . . . . . . . . . . . . . . . . . . 51 1.6 What is Reality? . . . . . . . . . . . . . . . . . . . . . . . . . 61 1.7 Some Results from Calculus . . . . . . . . . . . . . . . . . . . 71 2 Series 77 2.1 Introduction to Series . . . . . . . . . . . . . . . . . . . . . . . 77 2.2 Series with Nonnegative Terms . . . . . . . . . . . . . . . . . 85 2.3 Series with Terms of Both Signs . . . . . . . . . . . . . . . . . 91 2.4 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3 Sequences and Series of Functions 107 3.1 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . 107 3.2 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.3 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . 130 i

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Chapter 1 Sequences 1.1 The general concept of a sequence We begin by discussing the concept of a sequence. Intuitively, a sequence is an ordered list of objects or events. For instance, the sequence of events at a crime scene is important for understanding the nature of the crime. In this course we will be interested in sequences of a more mathematical nature; mostly we will be interested in sequences of numbers, but occasionally we will find it interesting to consider sequences of points in a plane or in space, or even sequences of sets. Let’s look at some examples of sequences. Example 1.1.1 Emily ﬂips a quarter five times, the sequence of coin tosses is HTTHT where H stands for “heads” and T stands for “tails”. As a side remark, we might notice that there are 2 5 = 32 different possible sequences of five coin tosses. Of these, 10 have two heads and three tails. Thus the probability that in a sequence of five coin tosses, two of them are heads and three are tails is 10 / 32, or 5 / 16. Many probabilistic questions involve studying sets of sequences such as these. Example 1.1.2 John picks colored marbles from a bag, first he picks a red marble, then a blue one, another blue one, a yellow one, a red one and finally a blue one. The sequence of marbles he has chosen could be represented by the symbols RBBYRB. 1

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2 CHAPTER 1. SEQUENCES Example 1.1.3 Harry the Hare set out to walk to the neighborhood grocery. In the first ten minutes he walked half way to the grocery. In the next ten minutes he walked half of the remaining distance, so now he was 3/4 of the way to the grocery. In the following ten minutes he walked half of the remaining distance again, so now he has managed to get 7/8 of the way to the grocery. This sequence of events continues for some time, so that his progress follows the pattern 1/2, 3/4, 7/8, 15/16, 31/32, and so on. After an hour he is 63/64 of the way to the grocery. After two hours he is 4095/4096 of the way to the grocery. If he was originally one mile from the grocery, he is now about 13 inches away from the grocery. If he keeps on at this rate will he ever get there? This brings up some pretty silly questions; For instance, if Harry is 1 inch from the grocery has he reached it yet? Of course if anybody manages to get within one inch of their goal we would usually say that they have reached it. On the other hand, in a race, if Harry is 1 inch behind Terry the Tortoise he has lost the race. In fact, at Harry’s rate of deceleration, it seems that it will take him forever to cross the finish line.
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SANDS10 - Sequences and Series An Introduction to...

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