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

$$ \frac12 + \frac14 + \frac18 + \frac1{16} + \cdots = 1 $$

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

$$ 1+2+3+ 4+5 + \cdots $$

With some infinite series, such as the harmonic series

$$ 1+\frac12 + \frac13 + \frac14 + \frac15 +\frac16 + \cdots $$

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

$$ a_1, a_2, a_3,\cdots, a_n,\cdots $$

in a given order. Each of $a_1,$ $a_2$, $a_3$ and so on represents a number. These are the terms of the sequence. For example the sequence

$$ 2, 3, 5, 7, 11, 13, \cdots $$

has first term $a_1 = 2$, second term $a_2 = 4$ and $n$th term $a_n$ is the $n$th prime number. The integer $n$ is called the index of $a_n$, indicating where $a_n$ 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

$$ 12, 14, 16, 18, 20, 22, \cdots $$

is described by the formula $a_n = 10 + 2n$. Or it can be described by the simpler formula $b_n = 2n$, 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 ${a_n}$. The order is important. The sequence $1,2,3,4,\cdots$ is not the same as the sequence $2,1,3,4,\cdots$

Ways to represent sequences

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

$$ a_n =\sqrt{n},\quad b_n = (-1)^{n+1} \frac1n,\quad c_n = \frac{n-1}{n},\quad d_n = (-1)^{n+1}. $$

or by listing terms,

$$ {a_n} = {\sqrt1, \sqrt 2, \sqrt 3,\cdots, \sqrt{n},\cdots} $$

$$ {b_n} = {1, -\frac12, \frac13, -\frac14,\cdots,(-1)^{n+1}\frac1n,\cdots} $$

$$ {c_n}={0, \frac12, \frac23,\frac34,\cdots, \frac{n-1}{n},\cdots} $$

$$ {d_n}={1,-1,1,-1,1,-1,\cdots, (-1)^{n+1},\cdots}. $$

Sometimes write

$$ {a_n} = {\sqrt{n}}_{n=1}^\infty. $$

• Two ways to represent sequences graphically.

Convergence and Divergence

Sometimes the number in the sequence approach a single value as the index $n$ increases, as with ${a_n = \frac1n}.$ In sequence ${c_n=\frac{n-1}{n}}$， terms approach $1$. In sequence ${d_n = (-1)^{n+1}}$ the sequence bounce back and forth between $1$ and $-1$, never converging to a single value.

Definitions (Converges, Diverges, Limit)

The sequence ${a_n}$ converges to the number $L$ if to every positive number $\epsilon$ there corresponds an integer $N$ such that for all $n>N$ it has

$$ |a_n-L|< \epsilon. $$

If no such number $L$ exists, we say that ${a_n}$ diverges.

If ${a_n}$ converges to $L$, we write $\lim_{n\to\infty}a_n= L,$ or simply $a_n\to L$, 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 $\infty$. We will exploit this connection to calculate limits of sequences.

Example 1. Applying the definition

Show that

1. $\lim_{n\to\infty}\frac1n = 0$
2. $\lim_{n\to \infty} c = c$, (for any constant sequence ${a_n \equiv c}$)

Example 2. A divergent sequence

Show that the sequence ${1,-1,1,-1,1,-1,\cdots}$ diverges.

Example 3. An unbounded sequence diverges

Show that the sequence ${\sqrt{n}}_{n=1}^\infty$ diverges.

Definition (Diverge to Infinity)

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

$$ \lim_{n\to \infty} a_n = \infty,\quad \text{or}\quad a_n\to \infty. $$

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

$$ \lim_{n\to \infty}a_n = -\infty,\quad \text{or} \quad a_n \to -\infty. $$

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

Calculating limits of Sequences

Theorem 1

Let ${a_n}$ and ${b_n}$ be sequences of real numbers and let $A$ and $B$ be real numbers. The following rules hold if $\lim_{n\to \infty} a_n = A$ and $\lim_{n\to \infty} b_n = B.$

1. Linear Rule: $\lim_{n\to\infty} (\alpha a_n\pm \beta b_n) = \alpha A \pm \beta B$, for constants $\alpha$ and $\beta$,
2. Product Rule: $\lim_{n\to\infty}(a_n\cdot b_n) = A\cdot B$,
3. Quotient Rule: $\lim_{n\to\infty} \frac{a_n}{b_n} = \frac{A}{B}$ if $B\neq 0$.

Example 3. Applying Theorem 1

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

1. $\lim_{n\to\infty}\left(-\frac{1}{n}\right) = - \lim_{n\to \infty}\frac{1}{n} = 0$
2. $\lim_{n\to\infty}\left(\frac{n-1}{n}\right) = \lim_{n\to\infty}\left(1-\frac{1}{n}\right) = \lim_n 1 - \lim_n \frac{1}{n} = 1.$
3. $\lim_{n\to\infty}\frac{4-7n^6}{n^6+3} = \lim_{n\to \infty}\frac{4/n^6 - 7}{1+3/n^6} = -7$

Remark It does not mean if the sum ${a_n+b_n}$ has a limit, then ${a_n}$ and ${b_n}$ both have limits.

Remark Every non-zero multiple of a divergent sequence ${a_n}$ diverges.

Theorem 2. The Sandwich Theorem for Sequences

Let ${a_n}$, ${b_n}$ and ${c_n}$ be sequences of real numbers. If $a_n\le b_n\le c_n$ holds for all $n$ beyond some index $N$, and if $\lim_{n\to\infty} a_n = \lim_{n\to\infty} c_n = L$, then $\lim_{n\to\infty}b_n = L$ as well.

Remark An immediate consequence of Theorem 2 is that, if $|b_n|\le c_n$ and $c_n\to 0$, then $b_n\to 0$ as well, because $-c_n \le b_n \le c_n$.

Example 4. Applying the Sandwich Theorem

Since $1/n\to 0$, we know that

1. $\frac{\cos n}{n} \to 0$
2. $1/2^n\to 0$
3. $(-1)^n \frac1n \to 0$

Theorem 3. The Continuous Function Theorem for Sequences

Let ${a_n}$ be a sequence of real numbers. If $a_n\to L$ and if $f$ is a function that is continuous at $L$ and defined at all ${a_n}$, then $f(a_n)\to f(L)$.

Example 5. Applying Theorem 3

Show that $\sqrt{n/(n+1)}\to 1$.

Example 6. The Sequence ${\sqrt[n]{k}}$

Show that $k^{1/n}\to 0$

Using l’Hôpital’s Rule

Theorem 4.

Suppose that $f(x)$ is a function defined for all $x\ge n_0$ and that ${a_n}$ is a sequence of real numbers such that $a_n = f(n)$ for $n\ge n_0$. Then

$$ \lim_{x\to\infty} f(x) = L\quad \implies \quad \lim_{n\to\infty} a_n =L. $$

Example 7. Applying L’Hôpital’s Rule

Show that $\lim_{n\to\infty} \frac{\ln n}{n} = 0$.

Example 8. Applying L’Hôpital’s Rule

Find $\lim_{n\to \infty} \frac{2^n}{5n}.$

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

Does the sequence $a_n = \left(\frac{n+1}{n-1}\right)^n$ converge?

Commonly Occurring Limits

Theorem 5.

The following six sequences converges to the limits listed below:

1. $\lim_{n\to \infty}\frac{\ln n}{n} = 0$
2. $\lim_{n\to\infty}\sqrt[n]{n} = 1$
3. $\lim_{n\to\infty}x^{1/n} = 1,\quad (x>0)$
4. $\lim_{n\to\infty}x^n = 0,\quad (|x|<1)$
5. $\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n = e^x,\quad (\forall, x)$
6. $\lim_{n\to\infty} \frac{x^n}{n!} = 0,\quad (\forall, x)$

Example 10. Applying Theorem 5

1. $\frac{\ln(n^2)}{n} \to ?$
2. $\sqrt[n]{n^2}\to ?$
3. $\sqrt[n]{3n}\to ?$
4. $\left(-\frac12\right)^n\to?$
5. $\left(\frac{n-2}{n}\right)^n\to ?$
6. $\frac{100^n}{n!}\to ?$

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. $a_1 = 1$ and $a_n = a_{n-1} + 1$
2. $a_1 = 1$ and $a_n = n\cdot a_{n-1}$
3. $a_1 = a_2 =1$ and $a_{n+1} = a_n + a_{n-1}$ (Fibonacci numbers)
4. $x_0=1$ and $x_{n+1} = x_n - \frac{\sin x_n - x_n^2}{\cos x_n - 2 x_n}$ defined a sequence that converges to a solution of the equation $\sin x - x^2 = 0.$

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 ${a_n}$ has the property that $a_n\le a_{n+1}$ for all $n$, then it is called a nondecreasing sequence.

Example 12. Nondecreasing Sequences

1. $1,2,3,\cdots, n,\cdots$
2. $\frac12, \frac23,\frac34,\cdots, \frac{n-1}{n},\cdots$
3. $3,3,3,\cdots, 3,\cdots$
4. $\frac21, (\frac32)^2, (\frac43)^3,\cdots, (\frac{n+1}{n})^n,\cdots$

Definition. Bounded , Upper Bound, Least Upper Bound

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

Example 13. Applying the Definition for Boundedness

1. The sequence $1,2,3,\cdots, n,\cdots$ has no upper bound.
2. $\frac12, \frac23,\frac34,\cdots, \frac{n}{n+1},\cdots$ 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 ${a_n}$ is a nondecreasing sequence with least upper bound $L$, then $\lim_{n\to\infty}a_n = L.$

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.