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
Other infinite series do not have a finite sum, as with
With some infinite series, such as the harmonic series
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
in a given order. Each of
has first term
Definition (Infinite Sequence)
An infinite sequence of numbers is a function whose domain is the set of positive integers.
Examples
- The sequence
is described by the formula
We denote a sequence by
Ways to represent sequences
- Sequence can be described by writing rules at specify their terms, such as
or by listing terms,
Sometimes write
- Two ways to represent sequences graphically.
Convergence and Divergence
Sometimes the number in the sequence approach a single value as the index
Definitions (Converges, Diverges, Limit)
The sequence
If no such number
If
**Remark**
The definition is very similar to the definition of the limit of a function
Example 1. Applying the definition
Show that
, (for any constant sequence )
Example 2. A divergent sequence
Show that the sequence
Example 3. An unbounded sequence diverges
Show that the sequence
Definition (Diverge to Infinity)
The sequence
Similarly if for every number
**Remark**
A sequence may diverge without diverging to infinity or negative infinity. As an example
Calculating limits of Sequences
Theorem 1
Let
- Linear Rule:
, for constants and , - Product Rule:
, - Quotient Rule:
if .
Example 3. Applying Theorem 1
By combining Theorem 1 with the limits of Example 1, we have
Remark
It does not mean if the sum
Remark
Every non-zero multiple of a divergent sequence
Theorem 2. The Sandwich Theorem for Sequences
Let
Remark
An immediate consequence of Theorem 2 is that, if
Example 4. Applying the Sandwich Theorem
Since
Theorem 3. The Continuous Function Theorem for Sequences
Let
Example 5. Applying Theorem 3
Show that
Example 6. The Sequence
Show that
Using l’Hôpital’s Rule
Theorem 4.
Suppose that
Example 7. Applying L’Hôpital’s Rule
Show that
Example 8. Applying L’Hôpital’s Rule
Find
Example 9. Applying L’Hôpital’s Rule to Determine Convergence
Does the sequence
Commonly Occurring Limits
Theorem 5.
The following six sequences converges to the limits listed below:
Example 10. Applying Theorem 5
Recursive Definitions
For the sequences that defined recursively by giving
- The value(s) of the initial term or terms, and
- A rule, called a recursion formula, for calculating any later term form terms that precede it.
Example 11. Sequence Constructed Recursively
and and and (Fibonacci numbers) and defined a sequence that converges to a solution of the equation
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
Example 12. Nondecreasing Sequences
Definition. Bounded , Upper Bound, Least Upper Bound
A sequence
Example 13. Applying the Definition for Boundedness
- The sequence
has no upper bound. is bounded above by
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
Proposition.
Suppose
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.