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Sequences And Series Notes Pdf

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Sequences and Series Class 11 Formulas & Notes

In fact, this chapter will deal almost exclusively with series. However, we also need to understand some of the basics of sequences in order to properly deal with series. We will therefore, spend a little time on sequences as well. To be honest, many students will never see series outside of their calculus class.

However, series do play an important role in the field of ordinary differential equations and without series large portions of the field of partial differential equations would not be possible. Sequences — In this section we define just what we mean by sequence in a math class and give the basic notation we will use with them. We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. More on Sequences — In this section we will continue examining sequences.

We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. Series — The Basics — In this section we will formally define an infinite series. We will also give many of the basic facts, properties and ways we can use to manipulate a series. We will also briefly discuss how to determine if an infinite series will converge or diverge a more in depth discussion of this topic will occur in the next section.

We will illustrate how partial sums are used to determine if an infinite series converges or diverges. We will also give the Divergence Test for series in this section. Special Series — In this section we will look at three series that either show up regularly or have some nice properties that we wish to discuss. Integral Test — In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges.

The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing. A proof of the Integral Test is also given. In order to use either test the terms of the infinite series must be positive. Proofs for both tests are also given. Alternating Series Test — In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. The Alternating Series Test can be used only if the terms of the series alternate in sign.

A proof of the Alternating Series Test is also given. Absolute Convergence — In this section we will have a brief discussion on absolute convergence and conditionally convergent and how they relate to convergence of infinite series. Ratio Test — In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges.

The Ratio Test can be used on any series, but unfortunately will not always yield a conclusive answer as to whether a series will converge absolutely or diverge. A proof of the Ratio Test is also given. Root Test — In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges.

The Root Test can be used on any series, but unfortunately will not always yield a conclusive answer as to whether a series will converge absolutely or diverge. A proof of the Root Test is also given. Strategy for Series — In this section we give a general set of guidelines for determining which test to use in determining if an infinite series will converge or diverge. A summary of all the various tests, as well as conditions that must be met to use them, we discussed in this chapter are also given in this section.

Estimating the Value of a Series — In this section we will discuss how the Integral Test, Comparison Test, Alternating Series Test and the Ratio Test can, on occasion, be used to estimating the value of an infinite series. Power Series — In this section we will give the definition of the power series as well as the definition of the radius of convergence and interval of convergence for a power series.

We will also illustrate how the Ratio Test and Root Test can be used to determine the radius and interval of convergence for a power series. Power Series and Functions — In this section we discuss how the formula for a convergent Geometric Series can be used to represent some functions as power series. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible.

However, use of this formula does quickly illustrate how functions can be represented as a power series. We also discuss differentiation and integration of power series. This will work for a much wider variety of function than the method discussed in the previous section at the expense of some often unpleasant work. Applications of Series — In this section we will take a quick look at a couple of applications of series.

We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. We will also see how we can use the first few terms of a power series to approximate a function.

Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width.

Class 11 Maths Revision Notes for Chapter-9 Sequences and Series

Infinite Sequence. Arithmetic Sequence or Progression. General term of Arithmetic Progression. Arithmetic Mean. Geometric Mean. Geometric progression.

diverge in the sense that neither converges to any number. Informally, a series is an expression consisting of numbers separated by plus signs, like. 1 + + ​.

Sequences and Series Class 11 Notes Maths Chapter 9

In fact, this chapter will deal almost exclusively with series. However, we also need to understand some of the basics of sequences in order to properly deal with series. We will therefore, spend a little time on sequences as well. To be honest, many students will never see series outside of their calculus class.

Sequence: Sequence is a function whose domain is a subset of natural numbers. It represents the images of 1, 2, 3,… ,n, as f 1 , f 2 , f 3 , …. Infinite Series: A series having infinite number of terms is called infinite series.

Important Topics Covered in JEE Main Sequences and Series Revision Notes

In JEE Main, JEE Advanced and other engineering entrances exams it is important for the candidate to remember all the important series and progressions. Thus, it is recommended that a serious candidate has a clear understanding of sequences and series. Arithmetic Progression AP is defined as a series in which a difference between any two consecutive terms is constant throughout the series. This constant difference is called the common difference. GP is defined as a series in which ration between any two consecutive terms is constant throughout the series.

Sequence A succession of numbers arranged in a definite order according to a given certain rule is called sequence. A sequence is either finite or infinite depending upon the number of terms in a sequence.

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1. Audomaro C.

27.04.2021 at 00:13