Sequence Definition and 1000 Threads

Homework Statement Let t_1=1, t_(n+1)= [1- 1/4(n^2)]*t_n for n>=1. The book says the limit is a Wallis product 2/pi, but I don't know where to start. I've been searching, but I'm lost. Could you point me in the right direction?

I've been struggling with this number sequence for some time now, and i can't find the pattern, can anyone help? the sequence is: 0, 15, 101, 8, 86, 9699, 6008, ... what comes after? any thoughts?

Homework Statement A sequence (Xn) is Cauchy if and only if, for every ε>0, there exists an open interval length ε that contains all except for finitely many terms of (Xn). Homework Equations The Cauchy Definition is: A sequence X = (xn) of real numbers is said to be a Cauchy sequence...

Homework Statement Show that the sum of a convrgent sequence and a divergent sequence must be a divergent sequence. What can you say about the sum of two divergent sequences? Homework Equations A theorem in the book states: Let {a_n} converge to a and {b_n} converge to b, then the...

NEVERMIND! IT IS 0! I SOMEHOW WAS STARING AT THE WRONG ANSWER SHEET FOR A LITTLE BIT! THANK YOU! 1. Homework Statement Determinte whether the sequence converges or diverges: (n^2)/(e^n)2. Homework Equations The book says that the solution is: e/(e-1). However, the limit of the equation...

Just an idea. which number is the next in sequence? 21, 34, 57, 61,...

Be {xn} a sequence that satisfies the condition 0 ≤ x_{m+n} ≤ x_{m} + x_{n}. Prove that lim_{n ->∞} xn/n exists. I'm kind of lost in this.

I feel like I'm missing something obvious, but anyway, in the text it states: lim as n→∞ of an+bn = ( lim as n→∞ of an ) + ( lim as n→∞ of bn ) But say an is 1/n and bn is n. Then the limit of the sum is n/n = 1, but the lim as n→∞ of bn doesn't exist and this property doesn't work...

Homework Statement f_{n} is is a sequence of functions in R, x\in [0,1] is f_{n} uniformly convergent? f = nx/1+n^{2}x^{2} Homework Equations uniform convergence \Leftrightarrow |f_{n}(x) - f(x)| < \epsilon \forall n>= n_{o} \inN The Attempt at a Solution lim f_{n} = lim...

Let <zn> be a sequence complex numbers for which Im(zn) is bounded below. Prove <e^(i*zn)> has a convergent subsequence. My question on this is what possible help could the boundedness of the Im(zn) to this proof and what theorem might be of help?

Homework Statement Show that a sequence ##f_n \to f \in C[0,1]## with the sup norm ##|| ||_\infty##, then ##f_n \to f \in C[0,1]## with the integral norm. The Attempt at a Solution given ##\epsilon > 0 \exists n_0 \in N## s.t ##||(fn-f) (x)|| < \epsilon \forall n > n_0## with ##...

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 \sum_{i=0}^{k} {k \choose i}f_{n+i} = f_{n+2k} Homework Equations Definition: f(0)=0 ; f(1)=1 f(n)=f_{n-1} + f_{n-2} for n>=2 The Attempt at a Solution I searched a basis for which the statement is true: n=2 and k=1 \sum_{i=0}^{1} {1 \choose i}f_{2+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 Prove that the sequence defined by x_1=3 and x_{n+1}= \frac{1}{4-x_n} The Attempt at a Solution Well I found like the first 4 terms of this sequence and it seems to be decreasing, heading closer to 0. So this sequence is probably bounded and if it decreasing then it...

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) <...