# Infinite series notes

Mathematically, a sequence is defined as a map whose domain is the set of natural numbers which may be finite or infinite and the range may be the set of real numbers or complex numbers. We can represent it as. A sequence is often confused with a set. Though they both appear to be same yet they are different.

A sequence is almost the same as a set except for the fact that in a set the elements cannot repeat while it is not so in case of a sequence. Moreover, there is no importance of order in a set, while the order matters a lot in sequence. The following example will further clear the difference between the two:. A sequence is said to be in Arithmetic Progression when they increase or decrease by a constant number.

This constant number is called the common difference d of the arithmetic progression. When three numbers are in A. Similarly we can find the two arithmetic means between two number. The concept of arithmetic mean has been already discussed in the previous sections.

This concept can be extended till the mth power for terms and such mean is termed as the arithmetic mean of mth power. Mathematically, the arithmetic mean for mth power is defined as. The simplest way to define a harmonic progression is that if the inverse of a sequence follows the rule of an arithmetic progression then it is said to be in harmonic progression.

The harmonic mean of two numbers is in fact the reciprocal of arithmetic mean of the reciprocal of the numbers. Another concept closely related to harmonic mean is that of weighted harmonic mean.

Harmonic mean is in fact a special case of weighted harmonic mean where all the weights are equal to 1 and is equal to any weighted harmonic mean having all equal weights. Let d be the common difference of the A. Since the terms within the brackets are either in an A.

Open System A system Skip to content. The value of the sum of n terms in a G. P depends on the value of r. We have different values according to whether r is equal to 1 or is not equal to 1. Harmonic Progression The simplest way to define a harmonic progression is that if the inverse of a sequence follows the rule of an arithmetic progression then it is said to be in harmonic progression.

Formula for Harmonic mean Let H be the harmonic mean between two numbers a and b. So, a, H, b are in H. Weighted Harmonic Mean Another concept closely related to harmonic mean is that of weighted harmonic mean.

Relation between A. Don't miss out! Subscribe To Our Newsletter. Learn new things. Get an article everyday. Give it a try.In this section we will introduce the topic that we will be discussing for the rest of this chapter. That topic is infinite series. So just what is an infinite series?

The most common names are : series notationsummation notationand sigma notation. You should have seen this notation, at least briefly, back when you saw the definition of a definite integral in Calculus I.

If you need a quick refresher on summation notation see the review of summation notation in the Calculus I notes. Now back to series. Had our original sequence started at 2 then our infinite series would also have started at 2. The infinite series will start at the same value that the sequence of terms as opposed to the sequence of partial sums starts. We do, however, always need to remind ourselves that we really do have a limit there!

We do have to be careful with this however. This section is going to be devoted mostly to notational issues as well as making sure we can do some basic manipulations with infinite series so we are ready for them when we need to be able to deal with them in later sections.

First, we should note that in most of this chapter we will refer to infinite series as simply series. So for example the following series are all the same. It is important to again note that the index will start at whatever value the sequence of series terms starts at and this can literally be anything.

Do not forget however, that there is a starting point and that this will be an infinite series. Now that some of the notational issues are out of the way we need to start thinking about various ways that we can manipulate series. We have the following properties. Yeah, it was just the multiplication of two polynomials. Each is a finite sum and so it makes the point. This is pretty much impossible since both series have an infinite set of terms in them, however the following formula can be used to determine the product of two series. The next topic that we need to discuss in this section is that of index shift. The basic idea behind index shifts is to start a series at a different value for whatever the reason and yes, there are legitimate reasons for doing that. Performing an index shift is a fairly simple process to do.Jason Starr. Back to Top. Definition, using the sequence of partial sums and the sequence of partial absolute sums. Absolutely convergent and conditionally convergent series are defined, with examples of the harmonic and alternating harmonic series.

Definitions of sequences and series, with examples of harmonic, geometric, and exponential series as well as a definition of convergence.

## List of Series Tests

Examples of the uses of manipulating or rearranging the terms of an absolutely convergent series. Daniel J. David Jerison. Five questions which involve finding whether a series converges or diverges, finding the sum of a series, finding a rational expression for an infinite decimal, and finding the total distance traveled by a ball as it bounces up and down repeatedly.

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Series, Convergence, Divergence. Select Sub-topic --select a subtopic-- Series, Convergence, Divergence. Sequences Definition, with examples of convergent and divergent sequences. Series Definition, using the sequence of partial sums and the sequence of partial absolute sums.

Introduction to Infinite Series Definitions of sequences and series, with examples of harmonic, geometric, and exponential series as well as a definition of convergence. Manipulating Absolutely Convergent Series Examples of the uses of manipulating or rearranging the terms of an absolutely convergent series.

Computing Series Partial Sums Steps for using a spreadsheet to compute the partial sums of a series. Infinite Series Complete exam problems 6C—1 to 6C—3 on page 41 Check solution to exam problems 6C—1 to 6C—3 on pages 94—6. Basic Definition of Infinite Series Five questions which involve finding whether a series converges or diverges, finding the sum of a series, finding a rational expression for an infinite decimal, and finding the total distance traveled by a ball as it bounces up and down repeatedly.Precalculus: Introduction to Sequences and Series.

Consider the natural numbers, a portion of which are shown below. This ordered group of numbers is an example of a sequence. More broadly, we can identify an arbitrary sequence using indexed variables:. The variables a i where i is the index are called terms of the sequence. Although this construct doesn't look much like a function, we can nevertheless define it as such: a sequence is a function with a domain consisting of the positive integers or the positive integers plus 0, if 0 is used as the first index value.

The range of this function is the values of all terms in the sequence. Coincidentally in the case of the natural numbers, the domain and range are identical assuming the first index value is an assumption that we will stick with here.

Also, we can reference the n th term of the sequence as just a n. Thus, for instance, given the sequence 1, 2, 3, 4, 5, 6, The sixth term, a 6is 6, for example. Consider the following example:. Practice Problem: Write the first five terms in the sequence. Solution: Remember that we are assuming the index n starts at 1. The terms are then. Series are similar to sequences, except they add terms instead of listing them as separate elements. A series has the following form. Again, we will assume that the first index of the series is 1 unless otherwise indicated 0 is the other common first index. We can express a series more succinctly using sum notation:. Note that in general, a series is infinite. This notation can also have the form below:. We can also consider a portion of the series: S nwhich is defined below, is called the n th partial sum of the series. As with sequences, series can be use algebraic expressions.

Consider the example below, which is the sum of all positive odd numbers. Note that the increment is 4 in each case. So, let's write the expression for the series as follows. This expression uses an initial index of 1, but you can also write an expression with an initial index of 0. Check the result by calculating the first several terms of the series. One of the most important questions we can ask about a sequence or series is whether it converges.

Otherwise, the sequence diverges. But this sounds very close to our definition of a limit. Although we will not cover the precise mathematical proof of convergence for a sequence, we can intuitively visualize what convergence looks like. Consider the following sequence. Let's graph the points of this sequence. Although the terms "oscillate" around 0, obviously they approach 0 as n approaches infinity. Try a very large value of n and plug it into the expression; you'll find the result is very close to zero.

This sequence converges. Furthermore, the limit of this sequence is 0, since. In a similar manner, a series converges if it is equal to a finite number. Otherwise, it diverges.This particular series is relatively harmless, and its value is precisely 1.

To see why this should be so, consider the partial sums formed by stopping after a finite number of terms. The more terms, the closer the partial sum is to 1. It can be made as close to 1 as desired by including enough terms.

Moreover, 1 is the only number for which the above statements are true. It therefore makes sense to define the infinite sum to be exactly 1.

The figure illustrates this geometric series graphically by repeatedly bisecting a unit square. Instead, the conclusion is that infinite series do not always obey the traditional rules of algebrasuch as those that permit the arbitrary regrouping of terms. The difference between series 1 and 2 is clear from their partial sums. The partial sums of 1 get closer and closer to a single fixed value—namely, 1. The partial sums of 2 alternate between 0 and 1, so that the series never settles down.

A series that does settle down to some definite value, as more and more terms are added, is said to convergeand the value to which it converges is known as the limit of the partial sums; all other series are said to diverge.

All the great mathematicians who contributed to the development of calculus had an intuitive concept of limitsbut it was only with the work of the German mathematician Karl Weierstrass that a completely satisfactory formal definition of the limit of a sequence was obtained. Consider a sequence a n of real numbers, by which is meant an infinite list a 0a 1a 2…. Stated less formally, when n becomes large enough, a n can be made as close to a as desired.

Every number in the sequence is greater than zero, but, the farther along the sequence goes, the closer the numbers get to zero. For example, all terms from the 10th onward are less than or equal to 0. Terms smaller than 0.

### Infinite Series

First, it does not involve any mystical notion of infinitesimals; all quantities involved are ordinary real numbers. Second, it is precise; if a sequence possesses a limit, then there is exactly one real number that satisfies the Weierstrass definition. Finally, although the numbers in the sequence tend to the limit 0, they need not actually reach that value.

The same basic approach makes it possible to formalize the notion of continuity of a function. Intuitively, a function f t approaches a limit L as t approaches a value p if, whatever size error can be tolerated, f t differs from L by less than the tolerable error for all t sufficiently close to p. Having defined the notion of limit in this contextit is straightforward to define continuity of a function.

Continuous functions preserve limits; that is, a function f is continuous at a point p if the limit of f t as t approaches p is equal to f p. And f is continuous if it is continuous at every p for which f p is defined.

Intuitively, continuity means that small changes in t produce small changes in f t —there are no sudden jumps. Earlier, the real numbers were described as infinite decimals, although such a description makes no logical sense without the formal concept of a limit.

This is because an infinite decimal expansion such as 3. It turns out that the real numbers unlike, say, the rational numbers have important properties that correspond to intuitive notions of continuity. Moreover, it varies continuously with x. See Sidebar: Incommensurables. In effect, there are gaps in the system of rational numbers. By exploiting those gaps, continuously varying quantities can change sign without passing through zero. The real numbers fill in the gaps by providing additional numbers that are the limits of sequences of approximating rational numbers.

Formally, this feature of the real numbers is captured by the concept of completeness. One awkward aspect of the concept of the limit of a sequence a n is that it can sometimes be problematic to find what the limit a actually is. However, there is a closely related concept, attributable to the French mathematician Augustin-Louis Cauchyin which the limit need not be specified.

The intuitive idea is simple.Do at least Foley Math Site. Search this site. Chapter 5. Chapter 6. Chapter 7. Chapter 8. Chapter 9 Infinite Series. Chapter AP Practice. Unit 1: Equations and Inequalities. Unit 2 Linear Equations. Unit 3.

## Teacher Websites

Unit 4 Quadratics. Unit 5 Polynomials. Unit 6 Rational Exponents and Radical Functions. Chapter 7 Exponential and Logarithmic Functions. Chapter 8: Rational Functions. Unit 9 Trigonometry. Unit 10 Sequence and Series. Unit 11 Probability. Supplemental Material. Assignment Calendar. Key: Conditions for convergence. Blank conditions for convergence. Study Guide Solutions. Alternating Series Test Video. Ratio and Root Test Video. Tay lor Polynomial Video.

Power Series Video. Natalie Foley, Dec 18,AM. Natalie Foley, Dec 18,PM. Convergence Workhseet SG. Review B Key. Review C Key. Review C worksheet. Series Review Key 1. Series Review worksheet.We also show a proof using Algebra below.

9.3-Infinite series pp 9-10 class notes

We often use Sigma Notation for infinite series. Our example from above looks like:. Let's add the terms one at a time. When the "sum so far" approaches a finite value, the series is said to be " convergent ":.

When the difference between each term and the next is a constant, it is called an arithmetic series. When the ratio between each term and the next is a constant, it is called a geometric series.

This illustration may convince you that the tems converge on 1 3 :. Maybe you can try to prove it yourself?

Try pairing up each plus and minus pair, then look up above for a series that matches. Using integral calculus trust me that area is ln 2 :. The order of the terms can be very important! We can sometimes get weird results when we change their order. For example in an alternating series, what if we made all positive terms come first?

So be careful! There are other types of Infinite Series, and it is interesting and often challenging! Hide Ads About Ads. Infinite Series The sum of infinite terms that follow a rule. When we have an infinite sequence of values: 1 21 41 81 16The sums are heading towards a value 1 in this caseso this series is convergent.

The "sum so far" is called a partial sum. So, more formally, we say it is a convergent series when: "the sequence of partial sums has a finite limit. It goes up and down without settling towards some value, so it is divergent. How do we know? So the harmonic series must also be divergent.