Lecture on Sequences and the Squeeze Theorem

This lecture focuses on section 5.1, dealing with sequences and specifically introduces the Squeeze Theorem. It also examines how sequences behave through examples and definitions connected to convergence, divergence, geometric sequences, and bounded sequences.

Introduction to Sequences

The lecture commences by reaffirming previously covered material on sequences, limits, convergence, and divergence which will pave the way for discussing series in subsequent lectures. Understanding sequences is essential as they serve as the foundational building blocks for series.

Squeeze Theorem Explained

Definition of the Squeeze Theorem

The Squeeze Theorem facilitates the determination of limits of sequences. It essentially posits that if a sequence, denoted as $b_n$, is bounded above by a sequence $c_n$ and below by another sequence $a_n$, and the limits of both of these bounding sequences are identical, then the sequence $b_n$ will also converge to the same limit. The formal statement of the theorem is as follows:

• Let $a_n <br>ightarrow l$ and $c_n <br>ightarrow l$ as $n <br>ightarrow ext{infinity}$
    - If $a_n ext{ ≤ } b_n ext{ ≤ } c_n$ for all integers $n$ sufficiently large,
    - Then, $ext{limit} \big(b_n \big) = l$.

Explanation Using the Example of Cosine

An illustrative example of the Squeeze Theorem is given through the sequence $rac{ ext{cos}(n)}{n^2}$. Since $ext{cos}(n)$ oscillates between -1 and 1, we have:

$-1 ext{ ≤ } ext{cos}(n) ext{ ≤ } 1$

Dividing the entire inequality by $n^2$ (which is positive for all $n$), we retain the inequality structure:

$- rac{1}{n^2} ext{ ≤ } rac{ ext{cos}(n)}{n^2} ext{ ≤ } rac{1}{n^2}$

As $n$ approaches infinity, both bounds converge to 0:

$ext{limit} \big(- rac{1}{n^2}\big) = 0 ext{ and } ext{limit} \big( rac{1}{n^2}\big) = 0$

Thus, by the Squeeze Theorem, we conclude:

$ext{limit} \big( rac{ ext{cos}(n)}{n^2}\big) = 0$.

Second Example: Sequence $- rac{1}{2n}$

Exploring the Oscillating Nature

We evaluate the sequence $b_n = - rac{1}{2n}$. Listing terms yields:

• For $n = 1$: $- rac{1}{2}$

• For $n = 2$: $rac{1}{4}$

• For $n = 3$: $- rac{1}{8}$

• Continues similarly.

Observations on Behavior

The sequence oscillates between negative and positive, indicating it is an oscillating sequence. The absolute values diminish as n increases:

$\big|- rac{1}{2}\big|, \big| rac{1}{4}\big|, \big|- rac{1}{8}\big|, … <br>ightarrow 0$

Hence, the limits of the bounding sequences are:

• $- rac{1}{2n}$ and $rac{1}{2n}$ both converge to 0 as n approaches infinity.

Thus, by the Squeeze Theorem, $ext{limit} \big(b_n\big) = 0$.

Third Example: Sequence $rac{2n- ext{sin}(n)}{n}$

Structure Analysis

The next sequence examined is $a_n = rac{2n - ext{sin}(n)}{n}$. Rewriting simplifies:

$ext{limit} \big(a_n\big) = 2 - rac{ ext{sin}(n)}{n}$

Squeeze Theorem Application

Similar to previous examples, we note:

• For $ext{sin}(n)$, the bounds remain between -1 and 1. Dividing by $n$ yields:

$- rac{1}{n} ext{ ≤ } rac{ ext{sin}(n)}{n} ext{ ≤ } rac{1}{n}$

As n approaches infinity, both bounds converge to 0. Thus, we conclude:

• The limit of $rac{ ext{sin}(n)}{n}$ approaches 0.

So we find the original sequence limit becomes:
$ext{limit} \big(a_n\big) = 2 - 0 = 2$.

Introduction to Geometric Sequences

Characteristics of Geometric Sequences

Geometric sequences have the form $r^n$, where r is a constant. The convergence properties depend on the value of r:

• If |r| < 1, then $ext{limit} \big(r^n\big) = 0$.

• If $r = 1$, then $ext{limit} \big(r^n\big) = 1$.

• If |r| > 1, the sequence diverges to infinity.

Connection to Previous Examples

The sequence $- rac{1}{2^n}$ serves as a geometric sequence with the geometric ratio $r = - rac{1}{2}$, which is less than one in absolute value. Hence, it converges to 0.

Understanding Bounded Sequences

Defining Bounded Sequences

A sequence $a_n$ is considered bounded above if there exists a real number m such that:
$a_n ext{ ≤ } m ext{ for all positive integers } n$

Conversely, it is bounded below if there exists a real number m such that:
$m ext{ ≤ } a_n ext{ for all positive integers } n$

A sequence is termed bounded if both conditions are satisfied. If not, it is unbounded, which means that it diverges.

Theorem on Convergence of Bounded Sequences

The theorem states that if a sequence converges, it is necessarily bounded. However, the converse is not necessarily true; a sequence can be bounded without converging.

Definitions of Increasing and Decreasing Sequences

Increasing Sequences

A sequence $a_n$ is increasing if:
$a_n ext{ ≤ } a_{n+1} ext{ for all } n ext{ greater than or equal to some } n_0$

This indicates that subsequent terms in the sequence are greater than or equal to their predecessors.

Decreasing Sequences

Conversely, a sequence $a_n$ is decreasing if:
$a_n ext{ ≥ } a_{n+1} ext{ for all } n ext{ greater than or equal to some } n_0$

Monotonic Sequences

A monotonic sequence is one that is either entirely increasing or decreasing. If it is bounded, the conditions of monotonicity assure convergence.

Monotone Convergence Theorem

Theorem Details

If a sequence $a_n$ is bounded and monotonic, then it converges. This points to the significance of demonstrating both conditions to assert convergence.

Example of Convergence with Factorials

As an example, the sequence $rac{4n}{n!}$ is assessed. By examining consecutive terms, it becomes evident that initial increases fall into a pattern, mixing both increase and decrease over consecutive terms. However, it eventually shows monotonicity and decreases beyond a certain n.

To ascertain convergence, we check:

1. Compare terms to demonstrate monotonic behavior.

2. Prove bounding properties.

Thus, we conclude that the series converges. Notably, factorials grow significantly faster than polynomial expressions, leading to the sequence approaching 0.