Sequences

2: Sequences

Definition of a Limit (more precise)

Definition

A sequence \(\{a_n\}\) has the limit \(L\) if
for every \(\epsilon >0 \) \(~~ \exists ~~N \backepsilon \)
if \(~~n > N \)
then
\(~~ |a_n - L| < \epsilon \)

Properties of Limits of Sequences

These properties are the same as the properties of limits of functions

Theorem:

Assume that the sequence \(\{a_n\}\) has a limit \(A\) and the sequence \(\{b_n\}\) has a limit \(B\).

Then,

  • \(\lim_{n\to\infty}\limits (a_n\pm b_n) = A \pm B\)
  • \(\lim_{n\to\infty}\limits (c a_n) = cA\)
  • \(\lim_{n\to\infty}\limits (a_n b_n) = A B\)
  • \(\lim_{n\to\infty}\limits (\frac{a_n }{b_n}) = \frac AB\) provided \(B\neq 0\)

Example:

The sequence \(\left\lbrace a_n \right\rbrace=\left\lbrace \frac{4n+3}{n} \right\rbrace\) has the limit \(4\) and the sequence \(\left\lbrace b_n \right\rbrace=\left\lbrace \frac{2n}{8n-1} \right\rbrace \) has the limit \(\frac{1}{4}\)

  1. Find \(\lim_{n\to\infty}\limits \left(\frac{a_n }{b_n}\right) \)
  2. Find \(\lim_{n\to\infty}\limits (a_n b_n)\)
  3. Find \(\lim_{n\to\infty}\limits \left(\frac{4n+3}{3n} \right)\)

Limits of Sequences as Functions

Theorem:
Suppose \(f\) is a function such that \(f(n) = a_n\) for all positive integers \(n\).
If \(\lim_{x\to\infty}\limits f(x)=L\), then the limit of the sequence \(\{a_n\}\) is also \(L\).
Definition:

\(\lim_{n\to\infty} \limits a_n=\infty\) means that for every positive number \(M\) there is an integer \(N\) such that if \(n>N\) then \(a_n > M\)

Example:

Find the limit of the sequence \(\left \{\frac{\sin(n)}{n}\right \}\)

Two useful Theorems:

Squeeze Theorem

If \( a_n \leq b_n \leq c_n \) and \(\displaystyle \lim_{n\to \infty}{a_n} = \lim_{n\to \infty}{c_n} = L\), then \(\displaystyle \lim_{n\to \infty} b_n=L\)


Absolute Value Theorem

If \(\lim_{n\to \infty} |a_n| = 0\), then \(\lim_{n\to \infty} a_n = 0\)

Example:

Evaluate \(\displaystyle \lim_{n\to\infty} \frac{(-1)^n}{n}\) if it exists

\(\displaystyle \lim_{n\to\infty} \left \vert \frac{(-1)^n}{n} \right \vert = \lim_{n\to\infty} \frac{1}{n} =0 \) So

\(\displaystyle \lim_{n\to\infty} \frac{(-1)^n}{n} =0 \)

Example:

Use the squeeze theorem to evaluate \(\displaystyle \lim_{n\to\infty} \frac{\sin(n)}{n}\) if it exists

Note

The geometric sequence \(\{r^n\}\) is convergent if \(-1 \lt r \leq 1\) and divergent for all other values of \(r\).

Futhermore, \[ \lim_{n\to\infty} = \begin{cases}0 & \text{ if } -1\lt r \lt 1 \\ 1 & \text{ if } r = 1\end{cases} \]

Defnition

A sequence \(\{a_n\}\) is increasing if \(a_n \lt a_{n+1}\)

A sequence \(\{a_n\}\) is decreasing if \(a_n \gt a_{n+1}\)

A sequence \(\{a_n\}\) is monotonic if it is increasing or decreasing

Definition

A sequence \(\{a_n\}\) is bounded above if there exists a number \(M\) such that

\[ a_n \leq M ~~\forall~ n \geq 1\]

A sequence \(\{a_n\}\) is bounded above if there exists a number \(m\) such that

\[ a_n \geq m ~~\forall~ n \geq 1\]

A sequence \(\{a_n\}\) is bounded if it is bounded above and below

Theorem

Every bounded, monotonic sequence is convergent.

Example

Determine whether or not the sequence \(\{e^{1/n}\}\) converges.

Growth Rates of Functions

There exists an \(N\) such that if \(n>N\), the following is true

\[\ln^p(n) \lt n^q \lt n^q \ln^r(n) \lt n^{q+s} \lt b^n \lt n! \lt n^n \]

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By jkesler

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