SEQUENCE MATHEMATIQUES

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Sequences and Series: An Introduction to Mathematical Analysis

The distinguishing feature of an arithmetic sequence is that each term is the arithmetic mean of its neighbors i e an= (an−1+an+1)/2 (see exercise 12) 1 1 THE GENERAL CONCEPT OF A SEQUENCE 7 A geometric sequence has the form aarar2ar3 for some ﬁxed num- bers a and r

PDF	Mathematical Sequences In mathematics informally speaking a sequence is an ordered list of objects (or events) Like a set it contains members (also called elements or terms) The number of ordered elements (possibly infinite) is called the length of the sequence

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Massive open online calculus

Reading mathematics is not the same as reading a novel—it’s more fun and more interactive! To read mathematics you need a pen plenty of blank paper and the courage to write down everything—even “obvious” things As you read a math book you work along with me the author trying to anticipate my next thoughts repeating many of the same calculat

PDF	Math 226: Sequences and Series Lecture 1: Sequences The goal of this course is to understand series and their use in mathematics A series is an expression of the form a 1+a 2+a 3+a 4+··· obtained by adding together inﬁnitely many quantities (The ··· are meant to say “keep going without end ”)

PDF	MATH10242 Sequences and Series

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81 ARITHMETIC SEQUENCES & SERIES

There are two major types of sequence arithmetic and geometric This section will consider arithmetic sequences (also known as arithmetic progressions or simply A P) The characteristic of such a sequence is that there is a common difference between successive terms For example: 1 3 5 7 9 11

How many terms are in arithmetic sequence?
An arithmetic series has twenty terms. The first term is –50 and the last term is 83, find the sum of the series. Thirty numbers are in arithmetic sequence. The sum of the numbers is 270 and the last number is 38. What is the first number? How many terms of the arithmetic sequence: 2, 2.3, 2.6, 2.9, . . . must be taken before the terms exceed 100?
What are some examples of arithmetic sequences?
Here are some examples of arithmetic sequences, see if you can determine a and b in each case: 1, 2, 3, 4, 5, ... 2, 4, 6, 8, 10, ... 1, 4, 7, 10, 13, ... The distinguishing feature of an arithmetic sequence is that each term is the arithmetic mean of its neighbors, i.e. an = (an−1 + an+1)/2, (see exercise 12).
What is a sequence in math?
A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines ... In particular, sequences are the basis for series, which are important in differential equations and analysis.
What is the 14th term of an arithmetic sequence?
The 14th term of an arithmetic sequence is 100. If the first term is 9, find the common difference. 7. The 10th term of an arithmetic sequence is –40. If the first term is 5, find the common difference. 8. If n + 5, 2n + 1 and 4n – 3 are three consecutive terms of an arithmetic sequences, find n. 9.

How to read do mathematics

Reading mathematics is not the same as reading a novel—it’s more fun, and more interactive To read mathematics you need a pen, plenty of blank paper, and the courage to write down everything—even “obvious” things. As you read a math book, you work along with me, the author, trying to anticipate my next thoughts, repeating many of the same calculat

Acknowledgments

This text is a modification of David Guichard’s open-source calculus text which was itself a modification of notes written by Neal Koblitz at the University of Washington and includes exercises and examples from Elementary Calculus: An Approach Using Infinitesimals by H. Jerome Keisler. I am grateful to David Guichard for choosing a Creative Common

R N

integers. We’ll address the idea of a real-valued function “filling in” the gaps between the terms of a sequence when we look at graphs in Section 1.5. Warning A common misconception is to confuse the sequence with the rule for generating the sequence. The sequences (an) and (bn) given by the rules an = ( 1)n and bn = cos(π n) are, despite appearan

1.3 Examples

Mathematics proceeds, in part, by finding precise statements for everyday concepts. We have already done this for sequences when we found a precise definition (“function from to ”) for the everyday concept of “a list of real numbers.” But people.math.osu.edu

1.3.3 Triangular numbers

The sequence of triangular numbers (Tn) is a sequence of integers counting the number of dots in increasingly large “equilateral triangles” built from dots. The term Tn is the number of dots in a triangle with n dots to a side. There are a couple of ways of making this discussion more precise. Given an equilateral triangle with n dots to a side, ho

1.7 Helpful theorems about limits

Not surprisingly, the properties of limits of real functions translate into properties of sequences quite easily. people.math.osu.edu

1.8 Qualitative features of sequences

Sometimes we will not be able to determine the limit of a sequence, but we still would like to know whether or not it converges to some unspoken number. In many cases, we can determine whether a limit exists, without needing to—or without even being able to—compute that limit. Your first exposure to mathematics might have been about constructions;

1.8.1 Monotonicity

And sometimes we don’t even care about limits, but we’d simply like some terminol-ogy with which to describe features we might notice about sequences. Here is some of that terminology. For instance, how much money I have on day n is a sequence; I probably hope that sequence is an increasing sequence. Definition A sequence is called increasing (or s

2.5 A test for divergence

Usually, the sequence of partial sums sn = a0 + a1 + + an is harder to under-stand and analyze than the sequence of terms ak. It would be helpful if we could say something about the complicated sequence sn by studying the easier-to-understand sequence ak. Specifically, if the sequence sn converges, what can be said about the sequence ak? If adding

3 Convergence tests

It is generally quite difficult—indeed, often impossible—to determine the value of a series exactly. Even if we can’t compute the value of a series, in many cases it is possible to determine whether or not the series converges. We will spend most of our time on this problem. people.math.osu.edu

3.1.2 Practice

The ratio test is particularly useful for series involving the factorial function. people.math.osu.edu

3.2 Integral test

If all of the terms an in a series are non-negative, then clearly the sequence of partial sums sn is non-decreasing. This means that if we can show that the sequence of partial sums is bounded, the series must converge. We know that if the series converges, the terms an approach zero, but this does not mean that an an+1 for every n. Many useful and

3.2.5 Integrating for approximations

Since it is typically difficult to compute the value of a series exactly, a good approx-imation is frequently required. In a real sense, a good approximation is only as good as we know it is, that is, while an approximation may in fact be good, it is only valuable in practice if we can guarantee its accuracy to some degree1. This guarantee is usual

5 Another comparison test

We’ve covered a ton of material thus far in this course; there is one more comparison test that comes in quite handy—the Limit Comparison Test—which we will meet in Section 5.2. The purpose of this chapter, however, runs deeper than “just” another comparison test. The emphasis on series has been almost entirely on the question of their conver-gence

5.2 Limit comparison test

This test is usually called the limit comparison test, people.math.osu.edu

6.4 Calculus with power series

Now we know that some functions can be expressed as power series, which look like infinite polynomials. Since calculus, that is, computation of derivatives and antiderivatives, is easy for polynomials, the obvious question is whether the same is true for infinite series. The answer is yes 1 people.math.osu.edu

X X 1 xn = xn,

n=0 n=0 which we already knew—it is just the geometric series. So given a function f , we may be able to differentiate it around zero, spot a pattern, and thereby compute the Taylor series, but that does not mean we have found a series representation for f . Worse, we don’t even know if the series we wrote down converges anywhere, let alone converg

Review

These problems require various techniques, and are in no particular order. Many problems may be done in more than one way. I also encourage you to try your hand at crafting your own problems: mathematics is more than answer-getting—it is also about problem-posing Building your own problems will give you a strong sense of the limitations of the mac

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