Sequences Definition and 576 Threads

Let $X=R$ and ${d}_{1}\left(x,y\right)=\frac{1}{\eta}\left| x-y \right|$ $\eta\in \left(0,\infty\right)$ and ${d}_{2}\left(x,y\right)=\left| x-y \right|$..By using ${d}_{1}$ and ${d}_{2}$ please show that ${x}_{n}=\left(-1\right)^n$ is divergent and ${x}_{n}=\frac{1}{n}$ is convergent...

Homework Statement I have a question about sequences in Pascal. I have been wondering if there is a way to write more than one element of succession on the output at once, without using arrays or strings. I guess best to illustrate on an example. Let's take a program that should determine...

Take, for example, ##x_{n+1}=x_n+2+4n\text{ with }x_0=0##. How would you express this explicitly in terms of n? The only method I've thought of is to assume it's of the form ##x_n=an^2+bn+c## and then write out the first few terms ##\{x_0=0,x_1=2,x_2=8\}## to get a system of equations for the...

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...

How do i go about deriving a general eqn for the nth term of a sequence provided an eqn of the sum to the nth term is given in terms of n?

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 Well I am stuck with this math problem and i was wondering if you could help me...

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 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...

Homework Statement I have given the statements: ##a_{n}^2 \ge x## , ##a_{n+1} \le a_{n}## , ##x > 0## and ##\inf a_{n} > 0 ##. How to prove the following: ##\lim_{n \to \infty}a_{n}=\sqrt{x}##Homework Equations ##a_{n}^2 \ge x## , ##a_{n+1} \le a_{n}## , ##x > 0## and ##\inf a_{n} > 0 ##...

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...

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 $$C$$...

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?