# Sequence And Series Convergence And Divergence Tests Pdf

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Worksheets in this series are not tagged with a US grade level, as we rely on teachers to use their own judgment to find a level of difficulty.

In mathematics , the ratio test is a test or "criterion" for the convergence of a series. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test. The usual form of the test makes use of the limit.

## Convergent series

Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of all the sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section.

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. Practice Quick Nav Download.

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## 8.4: Convergence Tests - Comparison Test

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Login Sign up Search for courses, skills, and videos. Convergent and divergent sequences. Partial sums: formula for nth term from partial sum. Partial sums: term value from partial sum.

In mathematics , a series is the sum of the terms of an infinite sequence of numbers. The n th partial sum S n is the sum of the first n terms of the sequence; that is,. Any series that is not convergent is said to be divergent or to diverge. There are a number of methods of determining whether a series converges or diverges. Comparison test. Ratio test.

## Sequence convergence/divergence

Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of all the sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section. 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.

### Sequences: Convergence and Divergence

We have seen that the integral test allows us to determine the convergence or divergence of a series by comparing it to a related improper integral. In this section, we show how to use comparison tests to determine the convergence or divergence of a series by comparing it to a series whose convergence or divergence is known. Typically these tests are used to determine convergence of series that are similar to geometric series or p-series. In the preceding two sections, we discussed two large classes of series: geometric series and p-series.

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\$—Sequences & Series: Convergence & Divergence suffice. Geometric Series, nth Term Test for Divergence, and Telescoping Series.

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