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 \(\frac{1}{4}\)
- Find \(\lim_{n\to\infty}\limits \left(\frac{a_n }{b_n}\right) \)
- Find \(\lim_{n\to\infty}\limits (a_n b_n)\)
- 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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