In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences), or the set of the first n natural numbers (for a sequence of finite length n). Sequences are one type of indexed families as an indexed family is defined as a function which domain is called the index set, and the elements of the index set are the indices for the elements of the function image.
For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...).
The position of an element in a sequence is its rank or index; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on the context or a specific convention. In mathematical analysis, a sequence is often denoted by letters in the form of
a
n
{\displaystyle a_{n}}
,
b
n
{\displaystyle b_{n}}
and
c
n
{\displaystyle c_{n}}
, where the subscript n refers to the nth element of the sequence; for example, the nth element of the Fibonacci sequence
F
{\displaystyle F}
is generally denoted as
F
n
{\displaystyle F_{n}}
.
In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them in computer memory; infinite sequences are called streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.
Homework Statement
An = (n+3)^1/(n+3)
Converge or diverges? Find the limit of the convergent sequences.
Homework Equations
The Attempt at a Solution
I have the solution but I don't understand it. I'm looking to get it.
It shows me taking the limit as x->infinity of x^1/x and...
Homework Statement
Let (rn) be an enumeration of the set Q of all rational numbers. Show that there exists a subsequence (rnk) such that limk\rightarrow\infty rnk = +\infty
Homework Equations
The Attempt at a Solution
Im not sure how to even attack this
I have a test on this stuff and I'm confused about some things.
First, how do I show work to this problem?
I know that its absolute value diverge but I don't know how to show the work. Also not sure how to show the work for alternating series test. Not sure how to show that the it's...
I know that for any two real sequences x_n and y_n, we have
\liminf_{n\to \infty} x_n + \liminf_{n\to \infty} y_n \leq \liminf_{n\to \infty} (x_n + y_n).
I also know that, if one of the sequences converges, the inequality becomes equality. My question is this: If I've managed to show that...
Homework Statement
For the sequence (see picture), explore its monotones, define supermum and infimum, minimum and maximum, and find if the sequence is convergent. If the sequence is convergent, find form where on the terms of this sequence differ from the limit for less than ε = 0.01.
The...
Homework Statement
let V be the vector space consisting of all infinite real sequences. Show that the subset W consisting of all such sequences with only finitely many non-0 entities is a subspace of VThe Attempt at a Solution
Ok so i have to show 1.Closure under Addition,2. Closure under...
1. Homework Statement
Sorry, I posted this earlier but I had an error in my problem statement; please advise. Thank you.
If a_n diverges to +inf, b_n converge to M>0; prove a_n*b_n diverges to +inf
2. Homework Equations
3. The Attempt at a Solution
My attempt follows: I...
Homework Statement
If a_n diverges to +inf, b_n converge to 0; prove a_n*b_n diverges to +inf
Homework Equations
The Attempt at a Solution
My attempt follows: I seem to have trouble getting things in the right order, so I am trying to work on my technique, with your help. Also...
Hi there!
Homework Statement
Ok here is my problem concerning a sequence that is bounded and should have a limit.
\Large x\geq0 and \Large a_{0}>\sqrt{x}
The sequence \Large a_{n} is defined by \Large a_{n+1}=\frac{1}{2}(a_{n}+\frac{x}{a_{n}}) where \Large n\geq0
So the first question is...
Homework Statement
If the sequence {a_n} n=1 to infinity converges to (a) with a_n >0 show {sqrt(a_n)}
converges to sqrt(a)
Homework Equations
hint: conjigate first
The Attempt at a Solution
abs[ (a_n-a) / (sqrt(a_n)+sqrt(a) ) ] < epsilon
i do not own LATEX, yet.
hi
I was having difficulty with this problem in the book
If (1/n) is a sequence in R
which points (if any) will it converge (for every open set there is an integer N such that for all n>N 1/n is in that open set) to using the following topologies
(a) Discrete
(b) Indiscrete
(c) { A in X ...
Given two sequences (hyperreal numbers):
(2,1,2,1...) and (1,3,1,3...) how can I order these? They are not compliments and don't seem to fit into any of the possible orderings. It seems that intuitively the second is larger than the first.
Homework Statement
Suppose the infinite series \sum a_v is NOT absolutely convergent. Suppose it also has an infinite amount of positive and an infinite amount of negative terms.
Homework Equations
The Attempt at a Solution
Say we want to prove it converges by proving...
Homework Statement
Hi, there are two questions that I'm quite stuck with.
1.Find the number of terms in each of these geometric sequences.
a) 1,-2,4...1024
b) 54,18,6...2/27
Homework Equations
ar^n-1
The Attempt at a Solution
1. a) r= -2
1x-2^n-1 ?
b)...
I sometimes have difficulty knowing how to approach problems where you have to evaluate a limit of a sequence of (Riemmann) integrals. I know that when the functions converge uniformly you can bring the limit inside. But when there is not uniform convergence, I was wondering if there are any...
Homework Statement
Hi, I was trying to work out this question, but i kinda got stuck. Can anyone help me please?
Thanks
4. Find the number of terms in each of these geometric sequences.
2,10,50...1250
Homework Equations
ar^n-1
The Attempt at a Solution
1250=2x5^n-1
The definition of a sequence of real numbers is : a function from N to R.
What is the standard way to handle a sequence like (1/log n) where the first term is undefined? Do we instead write (1/log (n+1)) so that the first term is defined? Or leave the first term undefined?
The definition...
I'm taking an introductory class on analysis right now and I'm trying to get through the book that we are reading. I'm having difficulty understanding a park of it and was hoping someone could help me out. The part I'm reading about now is on null sequences:
Here's an excerpt. I'm having...
Suppose I give you a sequence of numbers such as 2, 4, 6, 8, 10... and ask you to find the next integer. You would probably tell me 12, because the sequence follows the rule 2n where n is the ordinal number. But if I told you the next number in the sequence is 42, your rule wouldn't work, and...
First, I'm sorry for my bad english.
Homework Statement
I need to disprove:
(x_n) \in \ell^2 is a Cauchy sequence, if \displaystyle \lim_{x \to \infty} d(x_n, x_{n+1})=0.
Homework Equations
Ok, sequence is Cauchy sequence if \exists n_0 \; \forall p,q>0 \; d(x_p,x_q) \rightarrow 0...
Let )<C<\infty and a,b \in \mathbb{R}. Also let
Lip_{C}\left(\left[a,b\right]\right) := \left\{f:\left[a,b\right]\rightarrow \mathbb{R} | \left|f(x) - f(y)\right| \leq C \left|x-y\right| \forall x,y \in \left[a,b\right]\right\}
.
Let \left(f_{n}\right) _{n \in \mathbb(N)} be a sequence of...
As far as I understand, a sequence converges if and only if it is Cauchy. So say for some sequence a_n and for all epsilon greater than zero we have |a_n - a_{n+1}| < \epsilon for large enough n.
We could then say a_n converges if and only if \lim_{n \rightarrow \infty} a_n - a_{n+1} = 0 ...
Homework Statement
Let C=\bigcupn=1\inftyCn where Cn=[1/n,3-(1/n)]
a) Find C in its simplest form.
b)Give a non-monotone sequence in C converging to 0.
Homework Equations
The Attempt at a Solution
For part a) i get C=[0,3]. Is this correct? I am not sure as to wether 0 and 3 are...
Sequences, Cumulative Sums, Partial Sums Plot --- Ti-89 Titanium
My proffessor just gave us all these packets for all these programs he wants us to put into our calculators. They are for a Partial Sums Plot, and a List of Cumulative Sums, nth terms and differences. They're all for the Ti-83...
Sequences / Real Analysis question
Homework Statement
a,b are the roots of the quadratic equation x2 - x + k = 0, where 0 < k < 1/4.
(Suppose a is the smaller root). Let h belong to (a,b). The sequence xn is defined by:
x_1 = h, x_{n+1} = x^2_n + k.
Prove that a < xn+1 < xn < b, and then...
Homework Statement
If Xn is bounded by 2, and |X_{n+2} - X_{n+1}| \leq \frac{|X^2_{n+1} - X^2_n|}{8} , prove that Xn is a convergent sequence.
Homework Equations
The Attempt at a Solution
I believe the solution lies in proving Xn a Cauchy sequence, but I'm not sure how to work it...
Homework Statement
We are given three contractions which generate the Sierpinski right triangle:
A0 = \frac{1}{2} <x , y>
A1 = \frac{1}{2} <x-1 , y> + <1 , 0>
A2 = \frac{1}{2} <x , y-1> + <0 , 1>
We are asked to find the points on which the sequence
(A2\circA1)n(<x0 , y0>) ) --...
Not quite sure how to approach this problem at all. We are given three contractions which generate the Sierpinski right triangle:
A0 = \frac{1}{2} <x , y>
A1 = \frac{1}{2} <x-1 , y> + <1 , 0>
A2 = \frac{1}{2} <x , y-1> + <0 , 1>
We are asked to find the point to which the sequence...
Homework Statement
Let \phi_n(x) be positive valued and continuous for all x in [-1,1] with:
\lim_{n\rightarrow\infty} \int_{-1}^{1} \phi_n(x) = 1
Suppose further that \{\phi_n(x)\} converges to 0 uniformly on the intervals [-1,-c] and [-c,1] for any c > 0
Let g be any...
Hi everyone!
I would like to solve some questions:
Classify up to isomorphism the four-sheeted normal coverings of a wedge of circles. describe them.
i tried to to this and it is my understanding that such four sheeted normal coverings have four vertices and there are loops at each of...
Arithmetic Sequences - PLEASE HELP!
I would really appreciate any help to figure out the following 4 questions:
1) The Sum of the first two terms of an arithmetic progression is 18 and the sum of the first four terms is 52. Find the sum of the First eight terms
2) An arithmetic Progression...
Hello All,
Well to make the Story short, I made a little bet in school that I could solve all the number sequences on a site, but now's the joke. I can't
Ye ye , stop laughing But really I can handle myself at numbers normally pretty well, but this seems out of my league
Can anyone solve...
About the proof of the homology of the Klein Bottle here: http://en.wikipedia.org/wiki/Mayer%E2%80%93Vietoris_sequence#Klein_bottle.
I do not see how the conclusion follows from the fact that we can "choose" {(1,0), (1,-1)} as a basis for Z²...
Abstract algebra question. Given the short exact sequence
$ 1 \longrightarrow N \longrightarrow^{\phi} G \longrightarrow^{\psi} H \longrightarrow 1 $
I need to show that given a mapping $ j: H \longrightarrow G, and $ \psi \circ j = Id_h $ (the identity on H), then $ G \cong N \times H. (The...
Hi,
I need to prove that any infinite subsequence {xnk}of a Cauchy sequence {xn}is a Cauchy sequence equivalent to {xn}.
My problem is that it seemed way too easy, so I'm concerned that I missed something. Please see the attachment for my solution, and let me know what you think.
Thanks.
I understand that the limit of the sum of two sequences equals the sum of the sequences' limis: \displaysyle \lim_{n\rightarrow\infty} (a_{n} + b_{n}) = \lim_{n\rightarrow\infty}a_{n} + \lim_{n\rightarrow\infty}b_{n}. Similar results consequenly hold for sums of three sequences, four sequences...
Homework Statement
assuming an and bn are cauchy, use a triangle inequality argument to show that cn=
| an-bn| is cauchy
Homework Equations
an is cauchy iff for all e>0, there is some natural N, m,n>=N-->|an-am|<e
The Attempt at a Solution
I am currently trying to work backwards on this one...
How do you know if a sequence converges or diverges based on taking the limit?
here's an example
f:= 3^n/n^3;
if i take the limit the sequence goes to infinity.
does it diverge becuase the limit is not zero or can the limit be something other than zero and it still converge?
Homework Statement
Let f be a continuous function on [0,1]. Prove that if
\int_{0}^{1} x^n f(x) dx = 0
for all even natural numbers n, then f(x) = 0 for all x \in [0,1] .
Homework Equations
The Attempt at a Solution
I'm pretty much stuck on this problem. All I know is...
Homework Statement
I need to find if sequences have limits
1. 1,i,-1,-i,1,i,-i,1...
2. 1,i/2,-1/3,-i/4,1/5...
3. (1+i)/2,...,[(1+i)/2]^n
4.3+4i/5,...[(3+4i)/5]^n
Homework Equations
The Attempt at a Solution
I say 1 and 2 don't have limits because of the sign changes.
I say 3...
Say there is a sequence of points: { x_k,y_k } that has a convergent subsequence:
{ {x_k_i,y_k_i}} } that converges to: (x_0,y_0) .
Sorry for poor latex, it should read "x sub k sub i"
Can I extrapolate the sequence {x_k_i} and say it converges to x_0 seperately?
The reason I ask...
Homework Statement
Consider the sequence {an} where a1 = sqrt(k), an+1 = sqrt(k + an), and k > 0.
a. Show that {an} is increasing and bounded.
b. Prove that the limit as n approaches infinity of an exists.
c. Find the limit as n approaches infinity of an.
The attempt at a solution...
Well, implicitly in a problem i came accros something that looks like it first requires to establish the following result:(to be more precise, the author uses the following result in the problem)
Let \{y_n\} be a sequence. If
\lim_{n\rightarrow \infty}\frac{y_{n+1}}{y_n}=0,=>...
I've been thinking about sequences of length 2n, with n 1's and n 0's. In particular, I'm interested in such sequences that also have the following property: reading the sequence from left to right, at no point before the end of the sequence is the number of 1's and 0's equal. For example, a...
Any guidance or worked solutions would be appreciated
Homework Statement
For a potato race, a straight line is marked on the ground from a point A, and points B,C,D,... are marked on the line so that AB = BC = CD = ... = 2 metres. A potato is placed at each of the points B,C,D,...
A...
[b]1. Homework Statement [/b
Let's s denote the collection of all sequences in lR, let m denote the
collection of all bounded sequences in lR, let c denote the collection
of all convergent sequences in lR, and let Co denote the collection of
all sequences...
Homework Statement
prove or refute:
if lim(a(2n)-a(n)=o , then a(n) is a cauchy sequence
Homework Equations
The Attempt at a Solution
I need to prove that for every m,n big enough a(m)-a(n)<epsilon
so I know for all m and n I can say m=l*n, lim(a(m)-a(n))=lim(a(n*l)-a(n*l/2)...