Sequence Definition and 1000 Threads

Given any irrational number c > 0, prove that there is a strictly decreasing sequence of rational numbers that converges to c.

Homework Statement Rigorously show that for all x>0, the limit of {x^{1/n}} is 1. Homework Equations The Attempt at a Solution |x^{1/n}-1|\leq\epsilon I'm not sure where to go from here...just looking for a little guidance.

Homework Statement Construct a sequence that has all rational numbers in it Homework Equations None. The Attempt at a Solution Here are my thoughts, though I have no solutions yet. If I construct a sequence Sn= n*sin(n)-1/n, will it work? Thanks guys!

Homework Statement Hey, so here is the problem: Suppose 0≤c<1 and let an = (1 + c)(1 + c2)...(1 + cn) for integer n≥1. Show that this sequence is convergent. Well I understand the basic concepts of proving convergence of sequences, but in class we've only ever done it with sequences where...

Homework Statement Show that the sequence (x_1,x_2,x_3,...) defined by; Let x_1=1 for each n \in \mathbb{N} x_{n+1}= \frac{x_n}{2}+1 x_2=\frac{3}{2} Show that this sequence is bounded above by 2; that is prove that x_n\leq2 for all n\in\mathbb{N} The Attempt at a...

Hi guys, Homework Statement I'm trying to calculate the limit of the following sequence: an:= (\frac{n^{2}+10}{n^{2}-5n})^{(3n^{2}+2)} 2. The attempt at a solution Ok so I got this so far: an =...

A metric space of equivalent Cauchy sequence classes (Z, rho) is defined using a metric of the sequence elements in the space (X,d), where d is from XX to R (real numbers). The metric of the sequence classes is rho = lim d(S, T), where S and T are the elements of the respective sequences. To...

A data file contains a sequence of 8-bit characters such that all 256 characters are about as common: the maximum character frequency is less than twice the minimum character frequency. Prove that Huffman coding in this case is not more efficient than using an ordinary 8-bit fixed-length code...

Homework Statement If \{a_{n}\}\in\mathbb{R} is Cauchy, \forall\epsilon>0,\exists a subsequence \{a_{k_{j}}\} so that |a_{k_{j}}-a_{k_{j+1}}|<\frac{\epsilon}{2^{j+1}}. The Attempt at a Solution Since \{a_{k_{j}}\} is Cauchy,\forall\epsilon>0,\exists N_{\epsilon} such that for j,j+1\geq...

Homework Statement Prove that if f(x) is continuous for 0<f(x)<1, then lim_{n->\infty}\frac{1}{n}[(n+1)(n+2)(n+3)...(2n)]^{\frac{1}{n}}=\frac{4}{e}. Homework Equationsf(x)=log(1+x) The Attempt at a Solution We know that...

Homework Statement V0=4 V_{n+1}=\sqrt{V_{n}^{2}+2n+3} Homework Equations Show that Un is an arithmetic sequence. The Attempt at a Solution I counted Vn and i found that it equals: V_{n}=\sqrt{(Vn+2)^{2}+2} what is there to do after this?

I've created two summations for my coursework, now I need to show whether or not the summations are finite or infinite. The 2 summations are very similar: With the n/2 removed it was easy enough to show the sum was equal to 1 [edit: I now realize I may have this wrong], now with the n/2 term...

Syntax for a sequence "in" a set X It is well-known shorthand to say that a sequence of real numbers x_n is a sequence "in R". (Of course we do not mean that the function x_n is an element of R). In such a case, is it permissible to replace the word "in" by the element symbol, or is this not...

Here is a good question for maths enthusiasts. I really find this sum very tough. Any help, advice or guidance for this question will be greatly appreciated. find the product upto n terms (1+(1/2^2)) . (1+(1/3^2)) . (1+(1/5^2)) . (1+(1/7^2)) . (1+(1/11^2))...upto n terms Where the nth...

I have that the sequence a_n=\{2-(-1)^n\} not converges. I must show this with the rigorous definition. I think use \exists{\epsilon>0}\forall{N\in\mathbb{N}}\exists{n\geq N}:|a_n-\ell|\geq\epsilon How i can continue?

Let f : [a,b] → [a,b] satisfy |f(x)-f(y)| ≤ λ|x-y| where 0<λ<1. Prove f is continuous. Choose any Xo ε [a,b] and for n ≥ 1 define X_n+1 = f(Xn). Prove that the sequence (Xn) is convergent and that its limit L is a 'fixed point' of f, namely f(L)=L

Let (Xn) be a sequence satisfying |Xn+1-Xn| ≤ λ^n r Where r>0 and λ lies between (0,1). Prove that (Xn) is a Cauchy sequence and so is convergent.

1.Problem Statement: If O is an open subset of ℝ does there exist a sequence in O that converges to x? Explain. 2.Relevant equations 3. The Attempt at a Solution So if I define a open subset of ℝ to be open if for all points x \in O there exists a ε-neighborhood _{V}ε (a)...

Homework Statement Does the sequence 1 + .4 + .16 +.064 Converge and where if so? So I started this problem out by making the a series equation Ʃ.4^n Where n=0 So since i believe this is the series equation for this sequence I now know that it does in fact converge because -1 < .4 ≤...

Homework Statement given: Vn+1=Vn*cos(pi/2^(n+2)) Homework Equations Write a sequence Vn in terms of n The Attempt at a Solution we substitute n with n-1 and we get: V(n-1)+1= Vn-1*cos(pi/2^(n-1+2)) Vn=Vn-1*cos(pi/2^(n+1) Is this correct?

Homework Statement find the sum of the series to 'n' terms (1^3 / 1 ) + (1^3 + 2^3 / 1+2 ) + (1^3 + 2^3 + 3^3 / 1+2+3 ) +... Homework Equations sigma n^2 = n(n+1)(2n+1) / 6 sigma n = n(n+1) / 2 The Attempt at a Solution numerator = 1^3 + 2^3 + ... n^3 denominator = 1+2+3+...n...

Homework Statement given: Vn+1=Vn*cos(pi/2^(n+2)) and Vn= pi/2 Homework Equations Write Vn in terms of nThe Attempt at a Solution how can i start this. I already know that -1<cosx<1 and I've already proved that this sequence is a negative sequence. Then our teacher said if its...

Homework Statement Suppose f is continuously differentiable, and a < b. The sequence is defined as follows: a_{n} = n\int_a^b \! f(x) \, \mathrm{d} x - n(\frac{b-a}{n})\displaystyle\sum\limits_{i=1}^n f(a + \frac{b-a}{n}i) The Attempt at a Solution I've been busting my *** with this one...

Homework Statement Question 1: Prove that the cardinality of R (the set of all real numbers)is the same as the cardinality of R-{0} by constructing a bijective function from R to R-{0} Question 2: Let A be the infinite sequence of binary numbers as follows: A={(a1,a2,a3...)|ai= 0or 1 for...

Certain and Curious Number Sequence (w/ primes) This is the number sequence a(x) whose output is determined by the greatest integer divisor n of any factorization of a with the additional rule that if an element is repeated in the factorization, the factorization must be thrown out. EX: a(56)...

For a subset C is a subset of R^d the following conditions are equivalent: (i) C is closed. (ii) For every sequence (x_n) is a subset of C which converges in R^d, the limit lim (as n goes to infinity) x_n = x must also lie in C. Using the above theorem for the Sequence Criterion for...

Homework Statement For x,y \in\mathbb{R} define a metric on \mathbb{R} by d_2(x,y) = |\tan^{-1}(x) - \tan^{-1}(y) | where \tan^{-1} is the principal branch of the inverse tangent, i.e. \tan^{-1} : \mathbb{R} \to (-\pi/2 ,\pi/2). If (x_n)_{n\in\mathbb{N}} is a sequence in \mathbb{R} and...

Homework Statement The sum of an infinite geometric sequence is 131/2, and the sum of the first three terms is 13. Find the first term. Homework Equations S∞ = a/(1-r) Sn = a-arn/(1-r) The Attempt at a Solution a/(1-r) = 131/2 a-ar3/(1-r) = 13 2a = 27-27r ...... 1 a-ar3 =...

Our 8th grade math counts team met today and I didnt know how to do this problem: The first three terms of an arithmetic sequence are p, 2p+6, and 5p-12. What is the 4th term of this sequence? Please explain how to do this. Arigato!

Homework Statement Given the following data, calculate the main sequence lifetime of the Sun (in years), assuming that all the initial mass is hydrogen and all of it is converted into helium. Mass of the Sun = M = 2x1030kg Luminosity of the Sun = L = 4x1032W Energy released in the...

Hi, Homework Statement V is a non-empty, upper bounded subset of R. Show that a sequence (v_n)_{n \geq 0} in V exists such that: 1)v_0 \leq v_1 \leq ... and 2) the limit of the sequence is sup V. (Hint: use a recursive construction) Homework Equations The Attempt at a Solution...

If an = \sqrt[n]{2^n+3^n} does the sequence converge? Prove your assertion. I have no idea where to start with this problem. It does have something to with \exists N such that n>N \Rightarrow |an - a| < ϵ for all ϵ>0 Can yall help me? It would be greatly appreciated.

Homework Statement Given that limit of s_{2n} is L and limit of s_{2n+1} is L, prove that lim s_{n} is also L. Homework EquationsThe Attempt at a Solution This seems very obvious: If the even terms of a sequence approach a number and the odd terms of that sequence approach the same number...

Homework Statement A bounded monotone sequence converges. Proof for bounded monotone increasing sequence and decreasing sequence. Does both them converges?Homework Equations So, I used the least upper bound and great lower bound to prove increasing sequence and decreasing sequence...

Homework Statement I already have the solutions, but I am not sure what the solutions are trying to say. http://img194.imageshack.us/img194/2595/unledlvc.jpg So in I don't understand this, we have n > \frac{1}{\epsilon} and If (and I am guessing we really want this to...

We're approaching the bicentennial anniversary of the largest earthquake and swarm east of the Rocky Mountains, at least in recent history. Coincidentally, Oklahoma has just had it's largest earthquake in the state's history. Bicentennial of the New Madrid Earthquake Sequence December...

Homework Statement Two consecutive numbers from 1,2,3...n are removed A.M of remaining numbers is 105/4. Find n and those numbers removed .Homework Equations Answer n = 50 those numbers are 7 and 8 The Attempt at a Solution I solved this question like a few weeks ago but now it escaped my...

I asked this question before but I totally misunderstood what it was asking. Basically, I need to find that there exists a sequence {a_k} such that it converges to x for some x in R. Since the real numbers are equivalence classes of convergent Cauchy sequences the result seems fairly obvious...

Hello was wondering if anyone could help me prove that: Suppose (sn) converges to s not equal to 0 and ( sntn) converges to L. Prove that (tn) converges

If 0<r<1 and |x(n+1) - x(n)| < r ^n for all n. Show that x(n) is Cauchy sequence. help please.

I've been trying to learn some maths by myself. A book I found starts with a section on limits. I feel that I have a decent understanding of what is written, but then, there are some problems given that I just can't figure out. I feel like I'm missing something basic. I'm not sure what I'm...

Im struggling with the concept of this basic sequence question. Let x(n) be a sequence such that lim(n->00) (nx(n)) = 0 i.e. it converges to zero... How could i show that there is an N s.t. for all n≥N : -1 < nx(n) < 1 Any tips would be great.. I don't want an answer.. I want to...

I finished exploring a family of functions fk(x) = {xksin(1/x) for x≠0 {0 for x=0 for an assignment in my Analysis course, and I'm supposed to determine the highest order derivative that exists and whether or not it's continuous...

Simple question just as the title says, but I can't remember or derive the solution for the life of me. I know that the answer is 0. I know why the answer is 0. But I need to know the mathematical derivation of the solution, and that's the part that I can't remember. So, to reiterate, how do you...

Homework Statement Show that 2^{n^{1001}} |a_{n} - a_{\infty}| \rightarrow 0 as n \rightarrow \infty. Here, an is defined recursively by a_{1} = 1, a_{n+1} = \frac{1}{2}(a_{n}+\frac{x}{a_{n}}). I already know that a_{\infty} = \sqrt{x}.Homework Equations We are given a hint to consider...

Homework Statement 5^2 - 5^3 + 5^4 - ... + (-1)^k*5^k whre k is an integer with k >= 2 Homework Equations The Attempt at a Solution I know (5^(k-1) - 5^2)/2 gives you the sum if they were all positive. I tried multiplying it by (-1)^k or something but that just changes the sign. I...

Homework Statement The following series is a telescopic series. Find the exact sum of the series by performing a partial fraction decomposition and generalizing the formula for the nth partial sum Sn. Homework Equations The Attempt at a Solution 3n + 2 = A(n+1)(n+2) + B(n)(n+2) + C(n)(n+1)...

[PLAIN]http://img805.imageshack.us/img805/1575/photo0138d.jpg Hi, on thursday, i have exam of advance calculus and i could not solve two problem in study sheet given by İnstructor. By 9 question, i prove by add an subtract XnY to |XnYn-XY| and i have found that |Xn(Yn-Y)+...

Homework Statement Use the definition of a limit to prove that lim [(1+an)-1] = 1/2 if lim an = 1. Homework Equations (\forall\epsilon>0)(\existsN\inN)(n\geqN \Rightarrow|an-L|<\epsilon) The Attempt at a Solution Let \epsilon be arbitrary. Since lim an exists, \existsN\inN such...

Since I don't know how to use latex I have posed my question in word file. Yours help is greatly appreciated.