Sequences Definition and 590 Threads

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?

Homework Statement Prove that an+2=an+1+an where a1=1 and a2=1 is monotonically increasing. Homework Equations A sequence is monotonically increasing if an+1≥an for all n\inN. The Attempt at a Solution Base cases: a1≤a2 because 1=1. a2≤a3 because 1<2. Am I supposed to prove...

Homework Statement an= (n/n+2)^n ANS: 1/e^2 The Attempt at a Solution I was told this was convergent and I need to find the limit of the sequence. How do I do this, as I seem to keep getting that this is divergent. Isn't it divergent to infinity? Or am I missing something?

Homework Statement an= (n/n+2)^n The Attempt at a Solution I was told this was convergent and I need to find the limit of the sequence. How do I do this, as I seem to keep getting that this is divergent. Isn't it divergent to infinity? Or am I missing something?

I am reading Dummit and Foote Section 10.5 Exact Sequences - Projective, Injective and Flat Modules. I need some help in understanding D&F's proof of Proposition 27, Section 10.5, page 386 (see attachment). Proposition 27 reads as follows: (see attachment)...

I am reading Dummit and Foote Section 10.5 Exact Sequences - Projective, Injective and Flat Modules. I need some help in understanding D&F's proof of Proposition 25, Section 10.5 (page 384) concerning split sequences. Proposition 25 and its proof are as follows...

I am reading Dummit and Foote Section 10.5 Exact Sequences - Projective, Injective and Flat Modules. I need help with some of the conclusions to Example 2, D&F Section 10.5, pages 379-380 - see attached. However, note that the question is essentially about isomorphisms. However, I would like...

I wish to make a post regarding exact sequences, but I need a way in Latex to form the common symbol of a mapping arrow with a symbol over it - for an example of the symbols I mean please see attached. Since such symbols are littered through texts dealing with exact sequences, I really need to...

Problem: Define $a_n=(1^2+2^2+ . . . +n^2)^n$ and $b_n=n^n(n!)^2$. Recall $n!$ is the product of the first n natural numbers. Then, (A)$a_n < b_n$ for all $n > 1$ (B)$a_n > b_n$ for all $n > 1$ (C)$a_n = b_n$ for infinitely many n (D)None of the above Attempt: The given sequence $a_n$ can be...

A limit of a sequence is definitely convergent if: If for any value of K there is an N sufficiently large that an > K for n > N, OR for any value of K there is an N sufficiently large that an<±K for n > N My only question is what exactly are K, N, an and n? What values are they? How would...

Homework Statement Please look over my work and tell me if I did something wrong. Suppose Bn is a divergent sequence with the limit +∞, and c is a constant. Prove: lim cBn -> ∞ = +∞ for c > 0 Homework Equations N/A The Attempt at a Solution lim Bn -> ∞ = means that for some value K >...

Hello everyone, I'm starting to study sequences. I'm on Stewart's Calculus textbook (single variable, 7th edition, for those who have it, on p. 693). Now, I'm at the part where the limit laws are "transferred" to sequences. (I'm sorry. I do not know how to code. I hope this is clear to...

Homework Statement . I am trying to solve two exercises about complex sequences: 1) Let ##\alpha \in \mathbb C##, ##|\alpha|<1##. Which is the limit ##\lim_{n \to \infty} \alpha^n##?, do the same for the case ##|\alpha|>1##. 2) Let ##\mathcal M## be the set of the complex numbers ##c## such...

0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0 is a short exact sequence if the image of any morphism is the kernel of the next morphism. Thus, the fact that we have the 0 elements at the two ends is said to imply the following: 1. The morphism between A and B is a monomorphism...

I remember when I took Calculus B in college. I had never learned any math by reverse engineering before, but when I got to sequences and series, the only way for me to learn how to do it was to reverse engineer it. I had to look up the answer in the back of the textbook, and then work...

Hello all, Well, it's my first post here but I'm not unfamiliar to the forum having read different posts and been intrigued by maths for a while. I finally decided that, after looking at different career options while buying Spivak's Calculus and playing with numbers, I should go into...

How does one calculate the number of terms in the sequence \sum\limits_{a=2}^k \sum\limits_{b=a}^k of 1/(a*b).

Homework Statement b_{k} = {4, 1, 1, 4} x[n] = 2u[n] Write your answer using scaled unit impulse sequences and scaled unit step sequences. Write explicitly. Homework Equations The Attempt at a Solution 4114 2222222... ------------ 8228 8228 8228 8228 8228 ... ------------...

Homework Statement I have two questions 1.Today,in my test paper,I got this sequence. Find the nth term formula 1,3,15,61,253 I didn't know how to start.This is clearly not an arithmetic or geometric sequence. Any help? 2. And is there any formula for finding the nth term of sequences...

Prove that if (a_n) is a null sequence and (b_n) is a bounded sequence then the sequence (a_nb_n) is null: from definitions if b_n is bounded then ## \exists H \in \mathbb{R} ## s.t. ## |b_n| \leq H ## if a_n is a null sequence it converges to 0 (from my book), i.e. given ## \epsilon ' > 0 ##...

I know that a metric space is complete if every Cauchy sequence converges that will surely designate compact metric spaces as complete spaces . I need to see examples of metric spaces which are not complete. Thanks in advance !

was looking at a proof of this here: http://gyazo.com/8e35dc1a651cec5948db1ab14df491f8 I have two questions, why do you set K = max of all the terms of the sequence plus the 1 + |A| term? Why do you need the absolute value of all the terms? i.e. why |a_1| instead of |a_1|?

Homework Statement Prove that: The sequence x_n \to x if and only if there is a M > 0 such that \forall \epsilon > 0 , \exists n_\epsilon \in \mathbb{N} and n\geq n_\epsilon we have | x_n - x | < \epsilon M Homework Equations The first implication "=>" is proved by choosing M = 1...

Given a sequence ## <x_n> ##, let ## <x_{n+1}> ## denote the sequence whose nth term for each ## n \in \mathbb{N} ## is ## x_{n+1} ##. Show that if ## <x_n> ## converges then ## < x_{n+1} ## converges and they have the same limit. my attempt thus far given ## \epsilon > 0 ## ##\exists N...

Homework Statement Q1 Are the following sequences divergent or convergent as n tends to infinity. a: \frac{5n+2}{n-1} b: tan^{-1}(n) c:\frac{2^n}{n!} Q2 Evaluate:... a: \sum_{n=1}^{\infty} 3^{\frac{n}{2}} b: \sum_{n=1}^{99} (-1)^n Q3 Find whether the following converge or diverge...

Hi, Let a(n) be a real sequence such that a(n+1)-a(n) tends to zero as n approaches ∞. must a(n) converge? Also an explanation would be great thank you. have been wondering about this

Let c_00 be the subspace of all sequences of complex numbers that are "eventually zero". i.e. for an element x∈c_00, ∃N∈N such that xn=0,∀n≥n. Let {e_i}, i∈N be the set where e_i is the sequence in c_00 given by (e_i)_n =1 if n=i and (e_i)_n=0 if n≠i. Show that (e_i), i∈N is a basis for...

Let A be a nonempty subset of R that is bounded above and let α=supA. Show that there exists a monotone increasing sequence {an} in A such that α=lim an. Can the sequence {an} be chosen to be strictly increasing?

Define $\{a_n\}$ is integer sequences (all term are integers) satisfy condition $a_n=a_{n-1}+\left\lfloor\dfrac{n^2-2n+2-a_{n-1}}{n}\right\rfloor $ for $n=1,2,...$ *note: $\left\lfloor x\right\rfloor$ is a greatest integer number less than or equal $x$ Find general term of sequences.

Homework Statement The function f is defined on a neighborhood N of \bar{x}. Show that \lim_{x \rightarrow \bar{x}} f(x) = L if and only if \lim_{n \rightarrow \infty} f(x_n) = L when \{x-n\} is a sequence of points in N with \lim_{n \rightarrow \infty} x_n = \bar{x} . Homework...

Let {An} = 2^n / n! is it convergent or divergent and why?

That seems like a valid argument for showing that \phi_n converges to f, but I'm not sure how to show it's increasing. And as far \psi_n, converges, well I imagine that I'd use a similar argument, but I'm still not sure how to show it's decreasing.

Homework Statement These trickly little buggers always seem to confuse me. I need to find out whether or not the sequence is increasing, decreasing or neither. An=(n!)2/(2n)! Homework Equations The Attempt at a Solution I'm pretty sure that it's a decreasing sequences but when I expand and...

I am reading Dummit and Foote Section 10.5 on Exact Sequences. I am trying to understand Example 1 as given at the bottom of page 381 and continued at the top of page 382 - please see attachment for the diagram and explanantion of the example. The example, as you can no doubt see, requires...

I am reading Dummit and Foote on Exact Sequences and some of the 'diagrams that commute' have vertical arrows. Can someone please help me with the LaTex for these diagrams. I have given an example in the attachment "Exact Sequences - Diagrams with Vertical Arrows" - where I also frame my...

In exact sequences, a convenient notation is to have a function symbol like on a longrightarrow. I have given an example of this in the attachment so it is clear what notation I am requesting help with - see attachment "Exact Sequences - Latex Question" I have also provided an...

In Dummit and Foote Section 10.5 Exact Sequences (see attachment) we read the following on page 379: "Note that any exact sequence can be written as a succession of short exact sequences since to say X \longrightarrow Y \longrightarrow Z [where the homomorphisms involved are as...

Hi all, I have an all time doubt here. We know that if r.v z = x + y where x and y are 2 random sequences having corresponding pdfs p(x) and p(y), the pdf of z, p(z) = convolution ( p(x),p(y) ). I have seen the derivation for the continuous case although not thorough how to prove it. I...

Hello! Unfortunately, I have not spent as much time as I should have on limits, or sequences, or their properties. In trying to work on a number theory math proof I have come across the following: I have an infinite sequence of numbers, all between 0 and 1 inclusive. I know that the limit of...

I need to find the value of the first term for this geometric series. Sn = 33 tn = 48 r = -2 I know that I have to take the formulas tn = t1 x r^(n-1), and Sn = [t1 x (r^n) - 1] / (r - 1), and isolate t1 for the first formula and then input that into the second, but I don't know the actual...

Lets say that we have two Cauchy sequences {fi} and {gi} such that the sequence {fi} converges to a limit F and the sequence {gi} converges to a limit G. Then it can easily be shown that the sequence defined by { d(fi, gi) } is also Cauchy. But is it true that this sequence, { d(fi, gi) }...

Alright, I need some help with this. an = \frac{1 - 5n^{4}}{n^{4} + 8n^{3}} To find the limit of convergence, use l'Hopital's Rule. The result will come out to L = -5 From my book, "The sequence {an} converges to the number L if for every positive number ε there corresponds an...

This paper is still subject to being edited and I am an amatuer mathematician, so there may be some mistakes, typos, and "amatuer" notation. I really cannot believe the result here, so I assume something may obviously be wrong yet, and I just cannot figure it out. I am a CPA, and fairly good...

Homework Statement Let (fn) be the sequence of functions defined on [0,1] by fn(x) = nxnlog(x) if x>0 and fn(0)=0. Each fn is continuous on [0,1]. Let a be in (0,1) and ha(y) = yay := yey log(a). Using l'Hospitals rule or otherwise, prove that limy->+∞ ha(y) = 0. Then considering...

Could someone confirm that I've answered this question right please \[ Prove\ using\ the\ sequence\ definition\ that\ f(x)=10x^2\ is\ continuous\ at\ x_0=0\\ I\ have:\ take\ any\ sequence\ x_n\ converging\ to\ 0.\ Then\ f(x_n)=10x_n^2\ converges\ to\ f(x_0)=10*0^2=0\ so\ it\ is\ continuous.\\...

Homework Statement The problem is Exercise 8 from Chapter 7 of Rudin. It can be seen here: http://grab.by/mGxY Homework Equations The Attempt at a Solution It seems quite obvious to see that because \sum\left|c_n\right| converges, f(x) will converge uniformly. However...

Homework Statement Have a few limits that I'm stuck on: a) lim n->infinity (n(n+1)^(n+1))/(n+2)^(n+2)) b) lim n->infinity (n^n/(n+3)^(n+1)) c) lim n->infinity n^(-1)^n I've tried my best to understand what to do solve these, but can't get it. We've been given answers to standard...