Sequences
- Page ID
- 219479
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Definition of a Sequence
A sequence is a list of numbers, or more formally, a function f(n) from the natural numbers to the real numbers.
We write
an
to mean the nth term of the sequence.
Example:
If
1
an =
n + 2
then we have
a1 = 1/3, a2 = 1/4, etc.
Exercise:
Write the general term an for the following sequences:
-
-1, 1, -1, 1, -1, 1, ...
-
1, 4, 9, 16, ...
-
1/2, -1/6, 1/24, -1/120, ...
-
1, 1/2, -1/4, -1/8, 1/16, 1/32, -1/64, -1/128, ...
The Limit of a Sequence
Consider the sequence
1 2 3 4
, , , , ...
2 3 4 5
We see that as n becomes large the numbers approach 1. In particular if any small error number e is given, we can find an N such that for n > N, |an -1| < e. We say that the limit of the sequence approaches 1
In general,
If an is a sequence that converges to a limit L then for any \(\epsilon\) > 0,
we can find an N such that for all n > N
|an - L| < \(\epsilon\)
If there is no such L then we say that the sequence diverges.
Theorem
|
Example: We find the limit of the sequence
2n + 1
an =
n - 3
by considering the function
2n + 1
f(n) =
n - 3
We note that as
\(n \to \infty\)
we get
\( \frac{\infty}{\infty}\)
hence we can use L'Hopital's Rule: Taking derivatives of the top and bottom, we have 2/1 hence the limit is 2.
The Squeeze Theorem
Suppose that
lim an = lim bn = L
and that there is an N such that for any n > N,
an < cn < bn
then
lim cn = L
Example
Show that
\( \lim\limits_{n \to \infty} \frac{sin(n)}{n!} = 0 \)
Note that
-1 sin n 1
< <
n n! n
both the left hand and right hand sides converge to 0 hence
\( \lim\limits_{n \to \infty} \frac{sin(n)}{n!} = 0 \)
Monotonic and Bounded Sequences
|
Definition of Monotonicity and Boundedness an > an+1 (an < an+1 ) for all n. A sequence is bounded from above (below) if there is a number M such that an < M ( an > M) for all n.
|
Example
Determine the montonicity and boundedness of the following sequences
-
cos(n)
-
1/n3
Solution
-
cos(n) is not monotonic since, for example
cos 1 > cos 2
and
cos 3 < cos 4
However, cos(n) is bounded above by 1 and below by -1 since
-1 < cos(n) < 1 -
1/n3 is monotonic since for n > 0,
(1/n3)' = -3/n4 < 0
so 1/n3 is monotonically decreasing.
Exercises:
Classify the monotonicity and boundedness of the following sequences:
-
an = sin(n)
-
an = 1/n
-
n + 1
an =
n + 2 -
an = n2 + 1
|
Theorem A bounded monotonic sequence converges |
Example
Show that that sequence
n
an =
en
converges.
Solution
If
x
f(x) = = xe-x
ex
Then
f '(x) = (1 - x) e-x
Which is always negative for x > 1. Hence an is monotonic.
for x > 0,
xe-x < 1
so we can conclude that the sequence converges.
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