周次: 11 日期: May 6, 2022 节次: 2

In this lecture, we will discuss series and sequences, these are new concept and very important ones. Before talking about series, we will talk about sequences.

Overview

In this chapter we study how to add together infinitely many numbers. This is a subject of the theory of infinite series.

It is clear that infinite sum sometimes have a finite value, as in

12+14+18+116+=1

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Other infinite series do not have a finite sum, as with

1+2+3+4+5+

With some infinite series, such as the harmonic series

1+12+13+14+15+16+

it is not obvious whether a finite sum exists.

To explore such problems, we need to exam the limit for infinite many numbers first. A list of infinite many numbers is called a sequence.

§ 11.1. Sequences

Definition and Examples

A sequence is a list of infinite many numbers

a1,a2,a3,,an,

in a given order. Each of a1, a2, a3 and so on represents a number. These are the terms of the sequence. For example the sequence

2,3,5,7,11,13,

has first term a1=2, second term a2=4 and nth term an is the nth prime number. The integer n is called the index of an, indicating where an occurs in the list. We can think of the sequence as a function of n.

Definition (Infinite Sequence)

An infinite sequence of numbers is a function whose domain is the set of positive integers.

Examples

  • The sequence

12,14,16,18,20,22,

is described by the formula an=10+2n. Or it can be described by the simpler formula bn=2n, where the index n starts at 6 and increases. To allow such simpler formulas, we let the first index of the sequence be any integer.

We denote a sequence by an. The order is important. The sequence 1,2,3,4, is not the same as the sequence 2,1,3,4,

Ways to represent sequences

  • Sequence can be described by writing rules at specify their terms, such as

an=n,bn=(1)n+11n,cn=n1n,dn=(1)n+1.

or by listing terms,

an=1,2,3,,n,

bn=1,12,13,14,,(1)n+11n,

cn=0,12,23,34,,n1n,

dn=1,1,1,1,1,1,,(1)n+1,.

Sometimes write

an=nn=1.

  • Two ways to represent sequences graphically.

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Convergence and Divergence

Sometimes the number in the sequence approach a single value as the index n increases, as with an=1n. In sequence cn=n1n, terms approach 1. In sequence dn=(1)n+1 the sequence bounce back and forth between 1 and 1, never converging to a single value.

Definitions (Converges, Diverges, Limit)

The sequence an converges to the number L if to every positive number ϵ there corresponds an integer N such that for all n>N it has

|anL|<ϵ.

If no such number L exists, we say that an diverges.

If an converges to L, we write limnan=L, or simply anL, and call L the limit of the sequence.

**Remark** The definition is very similar to the definition of the limit of a function f(x) as x approaches . We will exploit this connection to calculate limits of sequences.

Example 1. Applying the definition

Show that

  1. limn1n=0
  2. limnc=c, (for any constant sequence anc)

Example 2. A divergent sequence

Show that the sequence 1,1,1,1,1,1, diverges.

Example 3. An unbounded sequence diverges

Show that the sequence nn=1 diverges.

Definition (Diverge to Infinity)

The sequence an diverges to infinity if for every number M there is an integer N such that for all n larger than N, an>M. If this condition holds we write

limnan=,oran.

Similarly if for every number m there is an integer N such that for all n>N we have an<m, then we say an diverges to negative infinity and write

limnan=,oran.

**Remark** A sequence may diverge without diverging to infinity or negative infinity. As an example 1,0,2,0,3,0, is also divergence.

Calculating limits of Sequences

Theorem 1

Let an and bn be sequences of real numbers and let A and B be real numbers. The following rules hold if limnan=A and limnbn=B.

  1. Linear Rule: limn(αan±βbn)=αA±βB, for constants α and β,
  2. Product Rule: limn(anbn)=AB,
  3. Quotient Rule: limnanbn=AB if B0.

Example 3. Applying Theorem 1

By combining Theorem 1 with the limits of Example 1, we have

  1. limn(1n)=limn1n=0
  2. limn(n1n)=limn(11n)=limn1limn1n=1.
  3. limn47n6n6+3=limn4/n671+3/n6=7

Remark It does not mean if the sum an+bn has a limit, then an and bn both have limits.

Remark Every non-zero multiple of a divergent sequence an diverges.

Theorem 2. The Sandwich Theorem for Sequences

Let an, bn and cn be sequences of real numbers. If anbncn holds for all n beyond some index N, and if limnan=limncn=L, then limnbn=L as well.

Remark An immediate consequence of Theorem 2 is that, if |bn|cn and cn0, then bn0 as well, because cnbncn.

Example 4. Applying the Sandwich Theorem

Since 1/n0, we know that

  1. cosnn0
  2. 1/2n0
  3. (1)n1n0

Theorem 3. The Continuous Function Theorem for Sequences

Let an be a sequence of real numbers. If anL and if f is a function that is continuous at L and defined at all an, then f(an)f(L).

Example 5. Applying Theorem 3

Show that n/(n+1)1.

Example 6. The Sequence kn

Show that k1/n0

Using l’Hôpital’s Rule

Theorem 4.

Suppose that f(x) is a function defined for all xn0 and that an is a sequence of real numbers such that an=f(n) for nn0. Then

limxf(x)=Llimnan=L.

Example 7. Applying L’Hôpital’s Rule

Show that limnlnnn=0.

Example 8. Applying L’Hôpital’s Rule

Find limn2n5n.

Example 9. Applying L’Hôpital’s Rule to Determine Convergence

Does the sequence an=(n+1n1)n converge?

Commonly Occurring Limits

Theorem 5.

The following six sequences converges to the limits listed below:

  1. limnlnnn=0
  2. limnnn=1
  3. limnx1/n=1,(x>0)
  4. limnxn=0,(|x|<1)
  5. limn(1+xn)n=ex,(,x)
  6. limnxnn!=0,(,x)

Example 10. Applying Theorem 5

  1. ln(n2)n?
  2. n2n?
  3. 3nn?
  4. (12)n?
  5. (n2n)n?
  6. 100nn!?

Recursive Definitions

For the sequences that defined recursively by giving

  1. The value(s) of the initial term or terms, and
  2. A rule, called a recursion formula, for calculating any later term form terms that precede it.

Example 11. Sequence Constructed Recursively

  1. a1=1 and an=an1+1
  2. a1=1 and an=nan1
  3. a1=a2=1 and an+1=an+an1 (Fibonacci numbers)
  4. x0=1 and xn+1=xnsinxnxn2cosxn2xn defined a sequence that converges to a solution of the equation sinxx2=0.

Bounded Nondecreasing Sequences

An important special type of sequence is one for which each term is at least as large as its predecessor.

Definition. Nondecreasing Sequence

If a sequence an has the property that anan+1 for all n, then it is called a nondecreasing sequence.

Example 12. Nondecreasing Sequences

  1. 1,2,3,,n,
  2. 12,23,34,,n1n,
  3. 3,3,3,,3,
  4. 21,(32)2,(43)3,,(n+1n)n,

Definition. Bounded , Upper Bound, Least Upper Bound

A sequence an is bounded from above if there exists a number M such that anM for all n. The number M is an upper bound for an. If M is the minimum upper bound among all the upper bounds for an, then M is called least upper bound for an.

Example 13. Applying the Definition for Boundedness

  1. The sequence 1,2,3,,n, has no upper bound.
  2. 12,23,34,,nn+1, is bounded above by M=1.

A nondecreasing sequence that is bounded from above always has a least upper bound. The existence for the least upper bound is the property of completeness of the real number system. We will prove that if L is the least upper bound then the sequence converges to L.

Proposition.

Suppose an is a nondecreasing sequence with least upper bound L, then limnan=L.

Theorem 6. The Nondecreasing Sequence Theorem

A nondecreasing sequence of real numbers converges if and only if it is bounded from above. If a nondecreasing sequence converges, it converges to its least upper bound.

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