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.
Let $\left(X,d\right)$ be a metric space. Let $A,B,S,T: X\to X$ be mappings satisfying
1) $T\left(X\right)\subset A(X)$ and $S\left(X\right)\subset B(X)$
2) The pairs $(S,A)$ and $(T,B)$ are weakly compatible and
3) $d\left(Sx,Ty\right)\le...
I am reading the book "An Introduction to Rings and Modules with K-theory in View" by A.J. Berrick and M.E. Keating ... ...
I am currently focused on Chapter 3; Noetherian Rings and Polynomial Rings.
I need help with another aspect of the proof of Proposition 3.1.2.
The statement and proof of...
I am reading the book "An Introduction to Rings and Modules with K-theory in View" by A.J. Berrick and M.E. Keating ... ...
I am currently focused on Chapter 3; Noetherian Rings and Polynomial Rings.
I need help with the proof of Proposition 3.1.2.
The statement and proof of Proposition 3.1.2...
I tried the comparison test for one B but not sure if I am right. Think it could also be a ratio test because of the variable exponent. I'm lost totally lost on number one A. Also, I have the answer for the first part of three but don't know how to do the second part of it by comparing.
Thanks
It's awful, the reading of this first chapter is extremely boring, he appears only to cover divergence and convergence of series, should i skip this chapter?
Well, I tried to do something similar to what I was suggested to do in http://mathhelpboards.com/calculus-10/convergence-sequence-15868.html. So I took polar coordinates:
Using that: $$ \left\{\begin{matrix}
a_{n} = r_{n}\cos(\theta_{n}) \\
b_{n} = r_{n}\sin(\theta_{n})
\end{matrix}\right. $$...
Mod note: Moved from a homework section.
1. Homework Statement
this is my lecturer's notes, he says it is a divergent series, but this seems like an obvious convergent series to me..
could someone verify?
Homework Equations
https://www.dropbox.com/s/mc5rth0cgm94reg/incorrect maths.png?dl=0...
First, please take a look at http://www.purplemath.com/modules/nextnumb.htm (the second-order sequence problem)
http://www.purplemath.com/modules/nextnumb.htm :
"Since these values, the "second differences", are all the same value, then I can stop. It isn't important what the second difference...
http://en.wikipedia.org/wiki/Exact_sequence
Let's look at the following short exact sequence:
0-->A-->B-->C-->0.
Since the sequence is exact, the mapping from A-->B will be invective and the mapping B-->C will be subjective. The wikipedia article says that we can think of the mapping A-->B as a...
Hi,
Please help me with this question: Investigate the convergence of the sequence tanx;tan2x;tan3x;...;tannx for xE(-90;90 degrees). Steps to follow: Find common ratio. Draw the graph. For which values will x converge. Determine sum to infinity.
I did try to solve, but file type too...
Homework Statement
Let an → 2. Prove from first principles (i.e. give a direct ε-N proof) that an2 → 4.
Homework EquationsThe Attempt at a Solution
I have tried considering |an-2|2 and considering that |an2-4| = |(an+2)(an-2)| but I could not get either of these methods to work. Would someone...
Homework Statement
\ell is the set of sequences of real numbers where only a finite number of terms is non-zero, and the distance metric is d(x,y) = sup|x_n - y_n|, for all n in naural-numbers
then the sequence u_k = {1,\frac{1}{2},\frac{1}{3},...,\frac{1}{k}, 0,0,0...}
and...
Homework Statement
Let lim f(x)=a as x appr. infinity
Let xn be a sequence so that lim xn=infinity as n appr. infinity. Prove using definitions that then lim f(xn)=a as n appr. infinity.
Homework Equations
[/B]The Attempt at a Solution
I have had hard time trying to grasp how to begin with...
I tackle the following game analysis:
2 players, two 6-sided dice. Bigger sum of points win.
First roller has an advantage, as he wins even if 2nd player's dice sum equals to his.
As the game is played with doubling cube (potentially increasing the odds before any roll), I tried to enumerate...
Homework Statement
My professor wants us to program on Python, where we have a certain sequence, for example:
sequence = ("one", "two", "three", "four")
I need to replace one of the sequence elements (example: "one") with another element (example: instead of the word "one", I need to put...
I need some help in fully understanding Example 1, section 4.3 Cauchy Sequences, page 73 of Apostol, Mathematical Analysis.
Example 1, page 73 reads as follows:
https://www.physicsforums.com/attachments/3844
https://www.physicsforums.com/attachments/3845
In the above text, Apostol writes:
"...
Homework Statement
Homework Equations
Down
The Attempt at a Solution
As you see in the solution, I am confused as to why the sum of residues is required.
My question is the sum:
$$(4)\cdot\sum_{n=1}^{\infty} \frac{\coth(\pi n)}{n^3}$$
Question #1:
-Why is the beginning n=1 the residue...
I saw this method of calculating:
$$I = \int_{0}^{1} \log^2(1-x)\log^2(x) dx$$
http://math.stackexchange.com/questions/959701/evaluate-int1-0-log21-x-log2x-dx
Can you take a look at M.N.C.E.'s method?
I don't understand a few things.
Somehow he makes the relation...
I am reading Adhikari and Adhikari's (A&A) book, "Basic Modern Algebra with Applications".
I am currently focussed on Section 9.7 Exact Sequences. On page 387, A&A give Theorem 9.7.1.
A&A use symbol in the exact sequences that looks like an oh but I think it should be a zero. They continue...
I just uploaded a post on the class SE of short exact sequences in Mod_R ... and tried to use Latex Code to achieve the same fancy script for S and E as Paul E Bland used in his text "Rings and Their Modules" ... but seemingly did not achieve the right script for S (E seems OK, or at least...
I am reading Paul E. Bland's book, "Rings and Their Modules".
I am trying to understand Section 3.2 on exact sequences in \text{Mod}_R and need help with the notion of the class \mathscr{S} \mathscr{E} of short exact sequences in the category \text{Mod}_R .
The subsection of Bland on the...
I am reading Paul E. Bland's book, "Rings and Their Modules".
I am trying to understand Section 3.2 on exact sequences in Mod_R and need help with the proof of Proposition 3.2.7.
Proposition 3.2.7 and its proof read as follows:I am having trouble in understanding the proof that condition (2)...
I am reading Paul E. Bland's book, "Rings and Their Modules".
I am trying to understand Section 3.2 on exact sequences in Mod_R and need help with the proof of Proposition 3.2.6.
Proposition 3.2.6 and its proof read as follows:The part of the proof that perplexes me is the section of the proof...
First, this is not the same question as
https://www.physicsforums.com/threads/goedel-numbering-decoding.484898/
It concerns a different encoding procedure, hence a different decoding one.
My question concerns the argument in http://en.wikipedia.org/wiki/G%C3%B6del_numbering_for_sequences
for...
Homework Statement
Show that \sum_{k=1}^n \frac{(-1)^{k-1}}{k} \binom{n}{k} = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n-1} + \frac{1}{n} = \sum_{k=1}^n \frac{1}{k}
Homework EquationsThe Attempt at a Solution
Writing out few of the summands:
\frac{n!}{1\cdot 1!(n-1)!} - \frac{n!}{2\cdot...
My textbook reads :
The graph of a_n=\frac{n}{n+1} are approaching 1 as n becomes large . In fact the difference
1-\frac{n}{n+1}=\frac{1}{n+1} can be made as small as we like by taking n sufficently large. We indicate this by writing \lim_{n \to \infty} \frac{n}{n+1}=1
I don't understand where...
How to divide two sequences in Wolfam Mathematica? For example
f_n=\frac{1}{n}=1,\frac{1}{2},\frac{1}{3},... and g_n=n^2=1,4,9,...
I want to get h_n=1,\frac{1}{8},\frac{1}{27}...=\frac{f_n}{g_n}
How to do that in Wolfram Mathematica?
Hi! (Wave)
I am looking at the following exercise:
If $\{ x_n \}$ is a sequence of rationals, then this is a Cauchy sequence as for the p-norm, $| \cdot |_p$, if and only if :
$$\lim_{n \to +\infty} |x_{n+1}-x_n|_p=0$$
That's what I have tried:
$\lim_{n \to +\infty} |x_{n+1}-x_n|_p=0$ means...
Homework Statement
I want to prove that if X is a normed space, the following statements are equivalent.
(a) Every Cauchy sequence in X is convergent.
(b) Every absolutely convergent series in X is convergent.
I'm having difficulties with the implication (b) ⇒ (a).
Homework Equations
Only...
Homework Statement
Let V consist of all infinite sequences {xn} of real numbers for which the series summation xn2 converges. If x = {xn} and y = {yn} are two elements of V, define (x,y) = summation (n=1 to infinity) xnyn.
Prove that this series converges absolutely.
Homework Equations
The...
A metric space is considered complete if all Cauchy sequences converge within the metric space. I was just curious if you could have a case of a metric space that doesn't have any Cauchy sequences in it. Wouldn't it be complete by default?
When trying to think of a space with no cauchy...
Hello Forum,
I am familiar with the arithmetic sequence (the difference between one entry and the previous one is constant) and the geometric sequence ( the ratio between one entry and the previous one is constant).
are there any other important and simple sequences I should be aware of...
Hi,let:
0->A-> B -> 0
; A,B Z-modules, be a short exact sequence. It follows A is isomorphic with B.
. We have that tensor product is
right-exact , so that, for a ring R:
0-> A(x)R-> B(x)R ->0
is also exact. STILL: are A(x)R , B(x)R isomorphic?
I suspect no, if R has torsion. Anyone...
Define
F(A) = The set of all finite subsets of A
Seq(A) = The set of all finite sequences with elements from A
Let A be an infinite set (not necessarily countable).
I want to prove the following lines.
1. Card seq(A) \le Card(A^\omega)
2. Card A = Card seq(A) = Card F(A)
Hi all
I'm studying sequences and series, the problem I am having is this:-
Given the sequence 8 11 14
Find the Nth Term.
I have worked out the nth term to be 3N+5, so if I wanted to find the 4th term it would be 3*4+5 = 17.
The problem is that I have come across the following...
I am reading Dummit and Foote, Section 10.5 : Exact Sequences - Projective, Injective and Flat Modules.
I am studying Proposition 29 (D&F, page 388)
I need some help in order to fully understand the proof of the last statement of Proposition 29.
Proposition 29 and its proof (Ch 10, D&F page...
I am reading Dummit and Foote, Section 10.5 : Exact Sequences - Projective, Injective and Flat Modules.
I am studying Proposition 28 (D&F pages 387 - 388)
I need some help in order to fully understand the proof of the last statement of Proposition 28.
Proposition 28 (Ch 10, D&F pages 387-388)...
I am reading Dummit and Foote, Chapter 10, Section 10.5, Exact Sequences - Projective, Injective and Flat Modules.
I need help with a minor step of D&F, Chapter 10, Theorem 28 on liftings of homomorphisms.
In the proof of the first part of the theorem (see image below) D&F make the following...
I am reading Dummit and Foote, Chapter 10, Section 10.5, Exact Sequences - Projective, Injective and Flat Modules.
I need help with a minor step of D&F, Chapter 10, Theorem 28 on liftings of homomorphisms.
In the proof of the first part of the theorem (see image below) D&F make the following...
Dummit and Foote open their section (part of section 10.5) on projective modules as follows:D&F then deal with the issue of obtaining a homomorphism from D to M given a homomorphism from D to L and then move to the more problematic issue of obtaining a homomorphism from D to M given a...
I am reading Dummit and Foote, CH 10 Section 10.5, Exact Sequences - Projective, Injective and Flat Modules.
As they introduce split sequences, D&F write the following:
I am concerned at the following statement:
"In this case the module B contains a sub-module C' isomorphic to C (namely C'...
Hi all,
I don't understand what is being asked by this question?
If anyone knows could they please describe the process, that would be greatly appreciated.
Homework Statement
I'm given a recursive sequence with the following initial terms:
##\begin{matrix}
f_0(0)=1&&&f_1(0)=0\\
f_0(1)=2&&&f_1(1)=1
\end{matrix}##
Now, I'm asked to justify that we have the following recursive relations:
##\begin{cases}
f_0(n)=2f_0(n-1)+f_1(n-1)\\...
It's been a while since I've dealt with sequences and series. Here is my explanation of sequences and series and let me know if I am right or wrong.
A sequence is just a list of numbers. By convention, we use the letter ##a## for sequences and they are written in a form like so...
Homework Statement So, I actually have a bunch of these problems and I cannot do any of them. I don't think I'm really understanding it. Here is the question: (one of them) The way I wrote them, a_n means a sub n
For each sequence a_n find a number k such that n^k a_n
has a finite non-zero...
We let C be the set of Cauchy sequences in \mathbb{Q} and define a relation \sim on C by (x_i) \sim (y_i) if and only if \lim_{n\to \infty}|x_n - y_n| = 0. Show that \sim is an equivalence relation on C.
We were given a hint to use subsequences, but I don't think they are really necessary...
Determine whether the sequence converges or diverges, if it converges fidn the limit.
a_n = n \sin(1/n)
so Can I just do this:
n * \sin(1/n) is indeterminate form
so i can use lopitals
so:
1 * \cos(1/x) = 1 * 1 = 1
converges to 1?