Sequences Definition and 576 Threads

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)...

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...

Show that (^{n}_{n}) - (^{n}_{n-1}) + (^{n}_{n-2}) - (^{n}_{n-3}) + ...(^{n}_{0}) = 0 (a+b)^{n} = \sum^{\infty}_{\nu=0} (^{n}_{\nu})a^{\nu}b^{n-\nu}a=1 b=-1 0 = (1+(-1))^{n} = \sum^{\infty}_{\nu=0}(^{n}_{\nu}) 1^{\nu}(-1)^{n-\nu} = \sum^{\infty}_{\nu=0}(^{n}_{\nu})(-1)^{n-\nu} = ...I don't...

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 / 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...

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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...

C(a,b) = a^2 + ab -b^2 The characteristic value of a Fibonacci sequence is an interesting property. 1) C(a,b) = C(a,a-b) 2) C(a,b) * C(c,d) = C(ac+ad-bd,bc-ad) 3) C(a,b) * C(a,a-b) = C(2a^2-2ab+b^2,2ab-b^2) =(C(a,b))^2 C(a,b)^n = C(A_{n},-B_{n})...

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)...

My textbook says the following: For a closed interval J_n = [a_n, b_n] "A nested sequence of rational intervals give rise to a separation of all rational numbers into three classes (A so-called Dedekind Cut). The first class consists of the rational numbers r lying to the left of the...

Is every sequence that converges a Cauchy sequence (in that for every e > 0, there is an integer N such that |a_n - a_m| < e whenever n,m > N)? I think it is because if a sequence a_n converges to L, then you can mark off an open interval of any size about L such that this interval contains all...

Homework Statement Prove that there exists two infinite sequences <an> and <bn> of positive integers such that the following conditions hold simultaneously: i) 1<a1<a2<a3...; ii) an<bn<(an)^2 for all n>=1 iii)(an) - 1 divides (bn) - 1 for all n>=1 iv)(an)^2 -1 divides (bn)^2 - 1 for all...

By definition, a sequence a(n) has the Cauchy sequence if for eery E>0 ,there exist a natural number N such that Abs(a(n) - a(m) ) < E for all n, m > N Could anyone tell me what is a(m) ? is it a subsequence of a(n) , or could it be any other non related sequence ?

Homework Statement Find sequences {f_n} {g_n} which converge uniformly on some set E, but such that {f_n*g_n} does not converge uniformly on E. Homework Equations The Attempt at a Solution I looked at some sequences of functions known to be convergent but not uniformly convergent...

Hi guys. Homework Statement Given any two sequences of integers of length n, \left{\{a_k\right\}_{k=1}^n,\left{\{b_k\right\}_{k=1}^n,, where 1\le a_k,b_k\le n for all 1\le k\le n, show that there are subsequences of a_k,b_k such that the sum of the elements in each subsequence is equal...

Homework Statement Suppose that (x_n) is a properly divergent sequence, and suppose that (x_n) is unbounded above. Suppose that there exists a sequence (y_n) such that limit (x_n * y_n) exists. Prove that (y_n) ===> 0. Homework Equations (x_n) ===> 0 <====> (1/x_n) ===> 0...