Sequence Definition and 1000 Threads

Im trying to understand the following. We have l_1(R)=( x=x_n in l(R): summation from n=1 to infinity for absolute value of x_n). It says that this summation converges, but converges to what? Also , its says (1) is not in l_1(R) but 1/n^2 is. Can some one explain how these are so. This is...

Homework Statement Consider ##R^2## with the sup norm ##|| ||_∞## defined by ##||x||_\infty=sup(|x_1|,|x_2|)## for ##x = (x1, x2)##. Show that a sequence ##x^{n} \in (R^2, || ||_\infty)## where ##x^{n} =(x^ {n}_1, x^{n}_2) ## converges to ##x = (x_1, x_2) \in R^2##...

I left this following question from my excercises last, hoping that solving the others will give me an insight onto how to proceed. But I still don't have a plan on how to start it: Consider the sequence s_n = Ʃ (k=0 to n) (t_k * p^k) in Q(rationals) with the p-adic metric (p is prime); where...

I left this following question from my excercises last, hoping that solving the others will give me an insight onto how to proceed. But I still don't have a plan on how to start it: Consider the sequence s_n = Ʃ (k=0 to n) (t_k * p^k) in Q(rationals) with the p-adic metric (p is prime); where...

Does anyone know if a set of homogeneous polynomials forms a reduced Grobner basis, then they form a regular sequence in the polynomial ring? Any references? All the references that I have looked at (so far) have not related the two. If this is not true, can you give me a counterexample...

I have the following exercise: Let s_n be a cauchy sequence in Q(rationals) not converging to 0. Show that there exists an e(epsilon) >0 and a natural number N such that either for all n>N, s_n > e or for all n>N, -s_n >e. I know that since Q is not complete, we cannot assume that there...

This is the problem I'm trying to slove: Consider the sequence s_n = Sumation (from k=0 to n) p^k (i.e. s_n=p^0+p^1+p^2...+p^n) in Q(rationals) with the p-adic metric (p is prime). Show that s_n converges to a rational number.[/B] Now, I do get some intuition on showing that the...

I have just started my first real analysis course and we are using Spivak's Calculus. We have just started rigorous epsilon-N proofs of sequence convergence. I was trying to do some exercises from the textbook (chapter 22) but there doesn't seem to be any mention of epsilon-N in the solutions...

I've noticed lots of interesting properties of the patterns of numbers in the Fibbonacci sequence that can be expressed as the sum of two squares. In fact, it's what got me into number theory in the first place. There seem to be no two adjacent entries that are not the sum of two squares- and it...

Homework Statement Assume x_n and y_n are Cauchy sequences. Give a direct argument that x_n+y_n is Cauchy. That does not use the Cauchy criterion or the algebraic limit theorem. A sequence is Cauchy if for every \epsilon>0 there exists an N\in \mathbb{N} such that whenever...

Homework Statement Consider three sequences a(n), b(n) and s(n) such that an a(n) <= s(n) <= b(n) for all n and lim(a(n)) = lim(b(n)) = s. Prove that lim(s(n)) = s (n) is a subscript Homework Equations the book i am doing this problem from is elementary analysis, the theory of calculus...

if we have a sequence (n,1/n) , n E N , the sequence converges? lim n = infinite lim 1/n = 0 (1,1),(2,1/2),(3,1/3)...(n,1/n) it is convergent and divergent?!

G finite abelian group WTS: There exist sequence of subgroups {e} = Hr c ... c H1 c G such that Hi/Hi+1 is cyclic of prime order for all i. My original thought was to create Hi+1 by reducing the power of one of the generators of Hi by a prime p. Then the order of Hi/Hi+1 would be p, but...

My main interest is usually physics, however I have become interested in the Fibonacci sequence. I looked at the sequence in detail and found a few interesting patterns. The main pattern was a relationship of the way that the digits of numbers compounded, the exact pattern was: 7,5,5,4,5,5...

Homework Statement Use the Monotone convergence theorem to give a proof of the Nested interval property. Homework Equations Monotone convergence theorem: If a sequence is increasing or decreasing and bounded then it converges. Nested Interval property: If we have a closed interval [a,b] and we...

Let X_1, X_2, ... be a sequence of random variables and define Y_i = X_i/E[X_i]. Does the sequence Y_1, Y_2, ... always convergence to a random variable with mean 1?

Homework Statement Let f_n(x) be a sequence of functions that converges uniformly to f(x) on the interval [0, 1]. Show that the sequence e^{f_n(x)} also converges uniformly to e^{f(x)} on [0,1]. Homework Equations The definition of uniform convergence. The Attempt at a Solution I...

I'm not and expert in math and have only high school level math which has mostly faded from memory. I've been searching the net to see if there is an algorithm or something to derive a function from a given input and output sequence of numbers. I'm guessing you just have to do it by hand as one...

Homework Statement Assume a_n is a bounded sequence with the property that every convergent sub sequence of a_n converges to the same limit a. Show that a_n must converge to a. The Attempt at a Solution Could I do a proof by contradiction. And assume that a_n does not converge...

Hi All; I attach a pdf file on something I have been working on for some time. Any feedback would be appreciated. Regards Garbagebin

Homework Statement Posted this thread earlier but had mis read the given answer. please disregard older thread as I don't know how to delete it! Write down the condition for the numbers p, q, r to form an arithmetic sequence & geometric progression. Homework Equations \ a_n =...

Homework Statement Write down the condition for the numbers p, q, r to form an arithmetic sequence. Homework Equations The Attempt at a Solution Have no idea, but I looked at the answer and they have assigned each letter with a given value (number). How is this possible?

I have spent many hours today trying to determine at what distances certain substances will condense in protoplanetary disks around stars or temperatures different to that of our sun. I have come to the conclusion that the Inverse-Square Law is key to determining this. However, I am unsure as...

Homework Statement Let I=[a,b], f : I to R be continuous and suppose that f(x) >= 0 . If M = sup{f(x):x ε I} show that the sequence $$\left( \int_a^b (f(x))^n \, dx \right)^\frac{1}{n}$$ converges to M The Attempt at a Solution Where do I start? I'm thinking of having g_n(x)=...

Find $\displaystyle\lim_{n\to\infty}\frac{\sqrt[n]{n}}{\sqrt[n+1]{n+1}}$.

Hello everyone, I need some help with the following: I understand that by using xn = axn-1+b we can generate a full sequence of numbers. For example, if x1=ax0+b, then x2 = ax1+b = a2x0+ab+b, and so on and so forth to xn. I need help applying this same concept to complex numbers (a+bi). Is...

I'm confused as to what to expect when I take, for example, the Fourier transform of a sequence of 16 pulses of varying duty cycles, repeating. That is, after the 16th pulse, the entire sequence repeats. My confusion is in the interaction of the frequency components of each pulse within the...

Homework Statement Does nsin(2\pi en!) converge and if so what does it converge to. this is a sequence and n is a positive integer. The Attempt at a Solution My teacher gave us a hint and to write e in a Taylor series. nsin(2\pi (1+1+\frac{1}{2!}+\frac{1}{3!}...)n!) so we...

Homework Statement Show that \sqrt{2},\sqrt{2\sqrt{2}},\sqrt{2\sqrt{2\sqrt{2}}} converges and find the limit. The Attempt at a Solution I can write it also like this correct 2^{\frac{1}{2}},2^{\frac{1}{2}}2^{\frac{1}{4}},2^{\frac{1}{2}}2^{\frac{1}{4}}2^{\frac{1}{8}} so each time i...

Homework Statement Suppose we have a sequence {x} = {x_1, x_2, ...} and we know that \{x\}\in\ell^2, i.e. \sum^\infty x^2_n<\infty. Does it follow that there exists a K>0 such that x_n<K/n for all n? Homework Equations The converse is easy, \sum 1/n^2 = \pi^2/6, so there would be a finite...

Homework Statement Why does lim( (1+(1/n))^n ) = e?Homework Equations If a_n convergent to a, and b_n converges to b, then (a_n * b_n) converges to (a * b)The Attempt at a Solution The lim(1 + (1/n)) = 1. If you multiply (1 + (1/n)) by itself n-times, you get the equation (1 + (1/n))^n, so...

The limit of an "almost uniformly Cauchy" sequence of measurable functions I'm trying to understand the proof of theorem 2.4.3 in Friedman. I don't understand why f must be measurable. The "first part" of the corollary he's referring to says nothing more than that a pointwise limit of a...

Homework Statement Let (a_n)_{n\in\mathbb{N}} be a real sequence such that a_0\in(0,1) and a_{n+1}=a_n-a_n^2 Does \lim_{n\rightarrow\infty}na_n exist? If yes, calculate it. Homework Equations The Attempt at a Solution I have a solution but I'd like to see other solutions..

Homework Statement Show that if x_n is a convergent sequence, then the sequence given by that average values also converges to the same limit. y_n=\frac{x_1+x_2+x_3+...x_n}{n} The Attempt at a Solution Should I say that x_n converges to some number P. so now I need to show that y_n...

Homework Statement Find limit of sequence An=(1-1/3)(1-1/6)...(1-1/(n(n+1)/2)) Homework Equations The Attempt at a Solution I just found limit of 1-1/(n(n+1)/2) when n→∞,which is 1.Is that a proper solution?

Suppose $f(x)$ is continuous and decreasing on $[0, \infty]$, and $f(n)\to 0$. Define $\{a_n\}$ by $a_n=f(0)+f(1)+\ldots+f(n-1)-\int_0^n f(x)dx$ (a) Prove $\{a_n\}$ is a Cauchy sequence directly from the definition. (b) Evaluate $\lim a_n$ if $f(x)=e^{-x}$.

Homework Statement Show that the sequence of real numbers defined by x_{n + 1} = x_n + \frac{1}{x_n^2}, \, x_1 = 1 is not a Cauchy sequence. Homework Equations A sequence \{ p_n \} is Cauchy if and only if, for all \varepsilon > 0, there exists an N > 0 such that d(p_n, p_m) <...

Homework Statement For x_{n} given by the following formula, establish either the convergence or divergence of the sequence X = (x_{n}) x_{n} := (-1)^{n}n/(n+1)Homework Equations The Attempt at a Solution This is for my real analysis class. I tried to use the squeeze theorem, but didn't get...

Homework Statement The sequence is 1 1 1 1 5 5 5 5 1 1 1 1 I need to find the formula for the sequence. Homework Equations The Attempt at a Solution I had a previous problem that was similar. It was a sequence of 1 5 1 5 1 5. I managed to get it with the formula of...

Homework Statement Let [x] be the greatest integer ≤x. For example [\pi ]=3 and [3]=3 Find lim a_n and prove it. a) a_n=[\frac{1}{n}] b) a_n=[\frac{10+n}{2n}] The Attempt at a Solution for the first one it will converge to zero. so can I write \frac{1}{n}< \epsilon then...

Homework Statement Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit. a) lim \frac{1}{6n^2+1}=0 b) lim \frac{3n+1}{2n+5}=\frac{3}{2} c) lim \frac{2}{\sqrt{n+3}} = 0 The Attempt at a Solution A sequence a_n...

Homework Statement does the series e^n/2^n converge or diverge? does the series 2^n/e^n converge or diverge? The Attempt at a Solution I took lim→∞ e^x/2^x and am getting ∞, so it diverges, right? I also used L'Hopital's rule and got the same result. My prof hinted that I was...

Homework Statement I'm having an enormously hard time wrapping my head around the following definition, which is using some concepts that keep showing up in other definitions and theorems. I'll state the definition and then i'll ask about the parts that i don't understand: We say that...

Homework Statement If {S_n} is a sequence whose values lie inside an interval [a,b], prove {S_n/n} is convergent. We don't know Cauchy sequence yet. All we know is the definition of a bounded sequence, and convergence and divergence of sequences. Along with comparison tests and Squeeze...

Homework Statement If a sequence {xn} in ℝn satisfies that sum || xn - xn+1 || for n ≥ 1 is less than infinity, then show that the sequence is Cauchy. Homework Equations The triangle inequality? The Attempt at a Solution || xm - xn || ≤ || Ʃ (xi+1 - xi) from i=n to m-1|| Using...

Homework Statement An = (((-1)^(n-1))n)/(n^2 + 1) I need to know if it converges or diverges and if it converges the limit. Homework Equations The Attempt at a Solution I know it converges to 0. But I don't know how to show it when evaluating. I tried evaluation An| in the...

Homework Statement Let a_n = 1 + 1/(1*2) + 1/(2*3) + ... + 1/(n*[n+1]). Prove {a_n} is bounded above. Homework Equations 1/(2*3) = 1/2 - 1/3 The Attempt at a Solution I accidentally left my notebook at school and I have no idea how to do this without my class notes. The book...

Hi, I'm new to PhysicsForums. I am a sophomore in high school and currently in Geometry/Honors Chemistry. I am fascinated with astrophysics, and I really want to learn everything about it. I will be teaching all of this to myself, so I'd just like the sequence. I think it goes like this...

e_{n+1} = (e_n-2)/(e_n+4) Prove that {e_n} converges to 0 if (a) e_0 > -1 (b) -2 < e_0 < -1 PS: I haven't learned things like sup and inf yet, so please don't use them.

e_{n+1} = e_n/(e_n+2) If -1 < e_0 < 0, prove that the sequence {e_n} converges to 0. PS: I haven't learned things like sup and inf yet, so please don't use them.