It also explores particular types of sequence known. Reading and writingas you read and study the chapter, fill the journal with examples for each lesson. We will then define just what an infinite series is and discuss many of the basic concepts involved with series. We will discuss if a series will converge or diverge, including many of the tests that can be used to determine if a. An arithmetic series is the sum of an arithmetic sequence. A guide to number patterns, sequences and series teaching approach this series covers revision of linear number patterns, introduction to quadratic sequences and finding the nth term. As a function of q, this is the riemann zeta function. Pupils need to have a good understanding of all number patterns and simultaneous equations from grade 10. A sequence has the limit l and we write or if we can make the terms as close to l as we like by taking n sufficiently large.
A decimal representation of a number is an example of a series, the bracketing of a real number by closer and closer rational numbers gives us an example of a sequence. Ncert solutions for class 11 maths chapter 9 sequences and. Solutions to the exercises in this booklet are available at the website. Arithmetic and geometricprogressions mctyapgp20091 this unit introduces sequences and series, and gives some simple examples of each. If and l series is absolutely convergent, if l1 then the series is divergent, and if l1 then the test is inconclusive.
The marginal propensity to consume mpc is defined as the percentage of a dollar by which consumption increases when income rises by a dollar. Notes on infinite sequences and series 7 1 12 14 y1x 0 0. All sequences and series exercise questions with solutions to help you to revise complete syllabus and score more marks. Now lets look at some special sequences, and their rules. Sequences and series this is a wikipedia book, a collection of wikipedia articles that can be easily saved, imported by an external electronic rendering service, and ordered as a printed book. A series converges if the sequence of partial sums converges, and otherwise the series diverges. If a n b n for every n large enough, then the series x1 n1 a n and x1 n1 b n either both converge or both diverge. Exercises on these topics are on the following pages. Any finite series has a sum, but an infinite geometric series may or may not have a sum. Thus, the first term corresponds to n 1, the second to n 2, and so on. Sequences and series are most useful when there is a formula for their terms. Determine what type of sequence the following are and then complete the problem. A series is built from a sequence, but differs from it in that the terms are added together. Fibonacci numbers fibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1.
Leading to applying the properties of geometric sequences and series to functions that illustrate growth and decay 1 arithmetic sequence term of a sequence or series. Ncert solutions for class 11 maths chapter 9 sequences and series. Review problems involving sequences and series pgs. Introduction to series and sequences math 121 calculus ii d joyce, spring 20 the goal. The purpose of this chapter is to introduce a particular way of. Notes on infinite series pdf 61p currently this section contains no detailed description for the page, will update this page soon. For example, exponential growth is a growth pattern that is. Sequentially complete nonarchimedean ordered fields 36 9. Calculus 2 geometric series, pseries, ratio test, root test, alternating series, integral test duration. Sequences and series lecture notes introduction although much of the mathematics weve done in this course deals with algebra and graphing, many mathematicians would say that in general mathematics deals with patterns, whether theyre visual patterns or numerical patterns.
They cover the properties of the real numbers, sequences and series of real numbers, limits of functions, continuity, di erentiability, sequences and series of functions, and riemann integration. A geometric series is the sum of the terms of a geometric sequence. Miss mathematical induction sequences and series john j oconnor 200910. Chapter 11 sequences and series 577 sequences and seriesmake this foldable to help you organize your notes.
Arithmetic sequences and series algebra 2, sequences and. For a given power series, there are only three posibilities. When we use the phrase sum of a series, we will mean the number that results from adding the terms, the sum of the series is 16. Begin with one sheet of 11 by 17 paper and four sheets of notebook paper. A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence. Page 838 sigma notation college algebra wednesday april 5. We find the sum by adding the first, a 1 and last term, a n, divide by 2 in order to get the mean of the two values and then multiply by the number of values, n. Prerequisite skills to be successful in this chapter, youll need to master. If r 1 the sequence converges to 1 since every term is 1, and likewise if r 0 the sequence converges to 0.
Remember that we are assuming the index n starts at 1. We discuss whether a sequence converges or diverges, is increasing or decreasing, or if the sequence is bounded. The easiest way to get used to series notation is with an. Free pdf download of ncert solutions for class 11 maths chapter 9 sequences and series solved by expert teachers as per ncert cbse book guidelines. Consider the successive quotients that we obtain in the division of 10 by 3 at. This note covers the notions of convergence of sequences and series and the nature of the real numbers. Mathematical induction 2 sequences 9 series power series 22 taylor series 24 summary 29 mathematicians pictures 30 exercises on these topics are on the following pages. A power series is like a polynomial of in nite degree. An infinite series is the formal sum of the terms of an infinite sequence. In an arithmetic sequence the difference between one term and the next is a constant. Again, we will assume that the first index of the series is 1 unless otherwise indicated 0 is the other common first index. Powered by create your own unique website with customizable templates.
Lesson 123 infinite sequences and series 777 example 3. In progressions, we note that each term except the first progresses in a definite manner. Note that a series is the sum of a number of terms of a sequence. Chapter 6 sequences and series of real numbers we often use sequences and series of numbers without thinking about it. Series are similar to sequences, except they add terms instead of listing them as separate elements. Since this is a geometric series, you can find the sum of the first 100 terms by using the formula sn a 1 1 n a r 1r. A better method of describing a sequence is to state how to determine the nth term with an explicit formula. Intuitively, a sequence is an ordered list of objects or events. In addition to certain basic properties of convergent sequences, we also study divergent sequences and in particular, sequences that tend to positive or negative in. Btw, that pdf was made by a mse user, and unless you knew italian, this years edition is the first in english. Sequence and seriesdefinition, types, formulas and examples.
Its just two different sets of equations, but if youve got them at your fingertips click on thumbnail to the right to see a pdf of all the formulas. The main purpose of our study of series and sequences is to understand power series. To continue the sequence, we look for the previous two terms and add them together. A sequence can be thought of as a list of numbers written in a definite order. A similar integral test would show that the series converges when q 1, while it diverges when q. Represent the sum of a series, using sigma notation determine the sum of the first n terms of an arithmetic or geometric series pgs. Example 1 write the first three terms in each of the following sequences defined by the following. Sequences via lists the method of using a list to specify a sequence perhaps is the most tricky, since it requires us to look at a short piece of a sequence, and guess at the pattern or rule that is being used to produce the terms in the sequence. This unit introduces sequences and series, and gives some simple examples of each. An infinite series is the indicated sum of the terms of an infinite sequence.
Find the common difference or the common ratio and write the equation for the nth term. Build a sequence of numbers in the following fashion. One of my favorites is sadly long out of printits w. For help with downloading a wikipedia page as a pdf, see help. Most elementary analysis texts contains chapters on sequences and series, but most of the best sources on the subject are the older prerudin analysis texts. A series can be finite for example, it might only have 25 terms or infinite, and the notation needs to allow for both. These are some notes on introductory real analysis. They dont include multivariable calculus or contain any problem sets. It also explores particular types of sequence known as arithmetic progressions aps and geometric progressions gps, and the corresponding series. To find a rule for s n, you can write s n in two different ways and add the results.
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