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4: Infinite Sequences and Series - Mathematics


IN THIS CHAPTER we consider infinite sequences and series of constants and functions of a real variable.

  • SECTION 4.1 introduces infinite sequences of real numbers. The concept of a limit of a sequence is defined, as is the concept of divergence of a sequence to (pminfty). We discuss bounded sequences and monotonic sequences. The limit inferior and limit superior of a sequence are defined. We prove the Cauchy convergence criterion for sequences of real numbers.
  • SECTION 4.2 defines a subsequence of an infinite sequence. We show that if a sequence converges to a limit or diverges to (pminfty), then so do all subsequences of the sequence. Limit points and boundedness of a set of real numbers are discussed in terms of sequences of members of the set. Continuity and boundedness of a function are discussed in terms of the values of the function at sequences of points in its domain.
  • SECTION 4.3 introduces concepts of convergence and divergence to (pminfty) for infinite series of constants. We prove Cauchy’s convergence criterion for a series of constants. In connection with series of positive terms, we consider the comparison test, the integral test, the ratio test, and Raabe’s test. For general series, we consider absolute and conditional convergence, Dirichlet’s test, rearrangement of terms, and multiplication of one infinite series by another.
  • SECTION 4.4 deals with pointwise and uniform convergence of sequences and series of functions. Cauchy’s uniform convergence criteria for sequences and series are proved, as is Dirichlet’s test for uniform convergence of a series. We give sufficient conditions for the limit of a sequence of functions or the sum of an infinite series of functions to be continuous, integrable, or differentiable.
  • SECTION 4.5 considers power series. It is shown that a power series that converges on an open interval defines an infinitely differentiable function on that interval. We define the Taylor series of an infinitely differentiable function, and give sufficient conditions for the Taylor series to converge to the function on some interval. Arithmetic operations with power series are discussed.

Series (mathematics)

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position when he reaches this second position, the tortoise is at a third position, and so on. Zeno concluded that Achilles could never reach the tortoise, and thus that movement does not exist. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise.

In modern terminology, any (ordered) infinite sequence ( a 1 , a 2 , a 3 , … ) ,a_<2>,a_<3>,ldots )> of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the ai one after the other. To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression like

The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the n th partial sums of the series. That is, [2]

Generally, the terms of a series come from a ring, often the field R > of the real numbers or the field C > of the complex numbers. In this case, the set of all series is itself a ring (and even an associative algebra), in which the addition consists of adding the series term by term, and the multiplication is the Cauchy product.


4: Infinite Sequences and Series - Mathematics

In this chapter we’ll be taking a look at sequences and (infinite) series. 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.

Series is one of those topics that many students don’t find all that useful. 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.

In other words, series is an important topic even if you won’t ever see any of the applications. Most of the applications are beyond the scope of most Calculus courses and tend to occur in classes that many students don’t take. So, as you go through this material keep in mind that these do have applications even if we won’t really be covering many of them in this class.

Here is a list of topics in this chapter.

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. We will also give many of the basic facts and properties we’ll need as we work with sequences.

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. We will also determine a sequence is bounded below, bounded above and/or bounded.

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).

Convergence/Divergence of Series – In this section we will discuss in greater detail the convergence and divergence of infinite series. 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. We will examine Geometric Series, Telescoping Series, and Harmonic Series.

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.

Comparison Test/Limit Comparison Test – In this section we will discuss using the Comparison Test and Limit Comparison Tests to determine if an infinite series converges or diverges. 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. Note as well that there really isn’t one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines. 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.

Taylor Series – In this section we will discuss how to find the Taylor/Maclaurin Series for a function. 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. We also derive some well known formulas for Taylor series of (<f e>^) , (cos(x)) and (sin(x)) around (x=0).

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.

Binomial Series – In this section we will give the Binomial Theorem and illustrate how it can be used to quickly expand terms in the form ( left(a+b ight)^) when (n) is an integer. In addition, when (n) is not an integer an extension to the Binomial Theorem can be used to give a power series representation of the term.


Infinite Geometric Sequence

A geometric sequence is one where the common ratio is constant an infinite geometric sequence is a geometric sequence with an infinite number of terms. For example:

  • 4, 12, 36 is a geometric sequence (each term is multiplied by 12, so r = 12),
  • 4, 12, 36,… is an infinite geometric sequence the three dots are called an ellipsis and mean “and so forth” or “etc. etc. etc.”

The standard proof is that this follows from the Taylor series

for the arctangent. This Taylor series is closely related to the Taylor series of the logarithm

and this is because the tangent function can be written in terms of complex exponentials, so the arctangent function can be written in terms of complex logarithms. So the appearance of $pi$ in this formula is morally due to Euler's formula.

But there is also the following beautiful proof, which I learned from Gabor Toth's Glimpses of Algebra and Geometry. Consider the number $N(r)$ of integer lattice points inside the circle of radius $r$ centered at the origin, or in other words the number of pairs of integers $x, y$ satisfying $x^2 + y^2 le r^2$. It is not hard to see that $N(r) sim pi r^2$ for large $r$ in fact, it is not hard to see that $N(r) = pi r^2 + O(r)$.

Let $r_2(n)$ denote the number of pairs of integers $(x, y)$ such that $x^2 + y^2 = n$. Then $N(r) = 1 + r_2(1) + . + r_2(r^2)$ (if $r$ is an integer). On the other hand, a classic result of number theory implies that

where $d_k(n)$ is the number of divisors of $n$ congruent to $k mod 4$. It follows that we can evaluate $N(r)$ by counting how many numbers between $1$ and $r^2$ are divisible by each number congruent to $1, 3 mod 4$ with the appropriate sign. This gives

$frac <4>= r^2 - leftlfloor frac <3> ight floor + leftlfloor frac <5> ight floor mp . $

and the result follows by taking the limit as $r o infty$.

For a historical perspective (i.e. if you want to see how geniuses struggled through things that would be natural today, mostly because of things they later discovered), the nice article The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha 1 by Ranjan Roy (1990) describes how the formula was discovered independently thrice:

The series (2) was obtained independently by Gottfried Wilhelm Leibniz (1646-1716), James Gregory (1638-1675) and an Indian mathematician of the fourteenth century or probably the fifteenth cen­tury whose identity is not definitely known. Usually ascribed to Nilakantha, the Indian proof of (2) ap­pears to date from the mid-fifteenth century and was a consequence of an effort to rectify the circle. […] Leibniz's work, in fact, was primarily concerned with quadrature the π/4 series resulted (in 1673) when he applied his method to the circle. Gregory, by comparison, was interested in finding an infinite series representation of any given func­tion and discovered the relationship between this and the successive derivatives of the given function. Gre­gory's discovery, made in 1671, is none other than the Taylor series note that Taylor was not born un­til 1685. […]

Finally, although the proofs of (2) by Leibniz, Gregory and Nilakantha are very different in approach and motivation, they all bear a relation to the modem proof given above.

Perhaps I'll come back and edit this post for a summary of their methods if I actually read the article. :-)

1 : Ranjan Roy, Mathematics Magazine, Vol. 63 (1990), pp. 291-306. I found this while flipping through the book Sherlock Holmes in Babylon: and other tales of mathematical history.

The infinite series $pi/4 = 1-1/3+1/5-1/7+ . $ can be established by finding the expression of Taylor series egin f(x) = sum_^infty frac<1> f^<(k)>(a) (x-a)^k end for $arctan(x)$ for $x in [-1,1]$ at $a = 0$ and applying the result for $x = 1$. The finite geometric sum formula egin sum_^n q^k = frac<1-q^><1-q>, q in mathbb, q eq 1 end is applied to find a Taylor-form series. The uniqueness of Taylor polynomial establishes the uniqueness of Taylor series. Note that we don't hence need to calculate all derivatives of $arctan(x)$. We calculate egin arctan(t) & = & arctan(t) - arctan(0) = iggvert_0^t arctan(x) = int_0^t frac<1> <1+x^2>dx & = & int_0^t Big(frac<1-(-x^2)^> <1-(-x^2)>+ frac<(-x^2)^> <1-(-x^2)>Big) dx & = & int_0^t frac<1-(-x^2)^> <1-(-x^2)>dx + int_0^t frac<(-x^2)^> <1-(-x^2)>dx & = & int_0^t sum_^n (-x^2)^k dx + int_0^t frac<(-x^2)^> <1+x^2>dx & = & sum_^n int_0^t (-x^2)^k dx + int_0^t frac<(-x^2)^> <1+x^2>dx & = & sum_^n int_0^t ((-1)x^2)^k dx + int_0^t frac<((-1)x^2)^> <1+x^2>dx & = & sum_^n int_0^t (-1)^k (x^2)^k dx + int_0^t frac<(-1)^(x^2)^> <1+x^2>dx & = & sum_^n int_0^t (-1)^k x^ <2k>dx + int_0^t frac<(-1)^x^<2(n+1)>> <1+x^2>dx & = & sum_^n (-1)^k int_0^t x^ <2k>dx + int_0^t frac<(-1)^x^<2n+2>> <1+x^2>dx & = & sum_^n (-1)^k iggvert_0^t frac> <2k+1>+ int_0^1 frac<(-1)^(tx)^<2n+2>> <1+(tx)^2>t dx & = & sum_^n (-1)^k frac> <2k+1>+ int_0^1 frac <(-1)^t^ <2n+2>x^<2n+2>> <1+(tx)^2>t dx & = & sum_^n frac<(-1)^k> <2k+1>t^ <2k+1>+ int_0^1 frac <(-1)^t^ <2n+3>x^<2n+2>> <1+(tx)^2>dx , end where $t in mathbb$ and $n in mathbb$. Note that $-x^2 eq 1$ for every $x in mathbb$. Hence we can apply the finite geometric sum formula for every $x in mathbb$, that allows us to calculate the Taylor polynomial for every $t in mathbb$. Assume now $t in [-1,1]$. To obtain the limit function we calculate egin Bigg| arctan(t) & - & sum_^n frac<(-1)^k> <2k+1>t^ <2k+1>Bigg| = Bigg| int_0^1 frac<(-1)^t^<2n+3>x^<2n+2>> <1+(tx)^2>dx Bigg| & leq & int_0^1 Bigg| frac<(-1)^t^<2n+3>x^<2n+2>> <1+(tx)^2>Bigg| dx & leq & int_0^1 frac<|-1|^|t|^<2n+3>|x|^<2n+2>> <|1+(tx)^2|>dx & = & int_0^1 frac <1^|t|^ <2n+3>x^<2n+2>> <1+(tx)^2>dx leq int_0^1 frac <|t|^<2n+3>x^<2n+2>> <1>dx & = & int_0^1 |t|^ <2n+3>x^ <2n+2>dx = |t|^ <2n+3>int_0^1 x^ <2n+2>dx & = & |t|^ <2n+3>iggvert_0^1 frac<1> <2n+3>x^ <2n+3>= frac<|t|^<2n+3>> <2n+3>leq frac<1^<2n+3>> <2n+3> & = & frac<1> <2n+3> ightarrow 0, end as $n ightarrow infty$. Hence egin arctan(x) & = & lim_ sum_^n frac<(-1)^k> <2k+1>x^ <2k+1>= sum_^infty frac<(-1)^k> <2k+1>x^ <2k+1>end for $x in [-1,1]$. Now inserting $x = 1$ into the series expression of $arctan(x)$ we obtain egin pi/4 & = & arctan(1) = sum_^infty frac<(-1)^k> <2k+1>1^ <2k+1>= sum_^infty frac<(-1)^k> <2k+1> & = & 1 - 1/3 + 1/5 - 1/7 + . , end that is the desired result. I hope that this was what you were searching for.


Infinite Series

Otherwise we say that the series (sumlimits_^infty <> ) diverges .

(N)th term test

If the series (sumlimits_^infty <> ) is convergent, then (limlimits_ = 0.)

Important!

The converse of this theorem is false. The convergence of (<>) to zero does not imply that the series (sumlimits_^infty <> ) converges. For example, the harmonic series (sumlimits_^infty ormalsize> ) diverges (see Example (3)), although (limlimits_ = 0.)

Equivalently, if (limlimits_ e 0) or this limit does not exist, then the series (sumlimits_^infty <> ) is divergent.

Properties of Convergent Series

Let (sumlimits_^infty <> = A ) and (sumlimits_^infty <> = B ) be convergent series and let (c) be a real number. Then


Infinite Sequences

A sequence of real numbers is a function (fleft( n ight),) whose domain is the set of positive integers. The values ( = fleft( n ight)) taken by the function are called the terms of the sequence.

The set of values ( = fleft( n ight)) is denoted by (left< <> ight>.)

A sequence (left< <> ight>) has the limit (L) if for every (varepsilon gt 0) there exists an integer (N gt 0) such that if (n ge N,) then (left| <– L> ight| le varepsilon .) In this case we write:

The sequence (left< <> ight>) has the limit (infty) if for every positive number (M) there is an integer (N gt 0) such that if (n ge N) then ( gt M.) In this case we write

If the limit (limlimits_ = L) exists and (L) is finite, we say that the sequence converges . Otherwise the sequence diverges .

Squeezing Theorem.

Suppose that (limlimits_ = limlimits_ = L) and (left< <> ight>) is a sequence such that ( le le ) for all (n gt N,) where (N) is a positive integer. Then

The sequence (left< <> ight>) is bounded if there is a number (M gt 0) such that (left| <> ight| le M) for every positive (n.)

Every convergent sequence is bounded. Every unbounded sequence is divergent.

The sequence (left< <> ight>) is monotone increasing if ( le <>>) for every (n ge 1.) Similarly, the sequence (left< <> ight>) is called monotone decreasing if ( ge <>>) for every (n ge 1.) The sequence (left< <> ight>) is called monotonic if it is either monotone increasing or monotone decreasing.


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":

12 + 14 + 18 + 116 + .

Term Sum so far
1/2 0.5
1/4 0.75
1/8 0.875
1/16 0.9375
1/32 0.96875
. .

The sums are heading towards a value (1 in this case), so 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."


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