Sequences Definition and 590 Threads

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...

Homework Statement Let X = (xn) be a sequence of positive real numbers such that lim(xn+1 / xn) = L > 1. Show that X is not a bounded sqeuence and hence is not convergent. Homework Equations Definition of convergence states that for every epsilon > 0 there exist some natural...

Homework Statement A sequence of terms U_{k} is defined by K \geq by the recurrence relation U_{k+2} = U_{k+1} - pU_{k} where P is a constant Given that U_{1} =2 and U_{2} = 4 a) find an expression in terms of p for U_{3} b) hence find an expression in terms of p for...

Hello all, I was wondering if you all could help me with a small problem. Me and a friend had a discussion about a number sequence i found on http://www.fibonicci.com/en/number-sequences 1, 3, 7, 11, 13 ... He says the next correct number is 27 and I say it's 17. This forum seems...

If there is a sequence of numbers 1,4,9,16... what is the 12th term in the sequence (where 1 is the first term). The textbook says that it is 144 since each term of the sequence is the square of the term value. I find (and have always found) this to be confusing, why can't an alternate algorithm...

Homework Statement For the sequence defined recursively as follows: a_1 = 2, and a_(n+1) = 1/ (a_n)^2 for all n from N. Homework Equations So, we are supposed to use induction to first fidn if the sequence increases or decreases, and then use induction again to show if it...

Homework Statement http://img394.imageshack.us/img394/5994/67110701dt0.png Homework Equations A banach space is a complete normed space which means that every Cauchy sequence converges. The Attempt at a Solution I'm stuck at exercise (c). Suppose (f_n)_n is a Cauchy sequence in E. Then...

In words, the sum of a geometric sequence can be written out to say "the first term divided by (1 minus the common ratio)". Does the first term also apply when the series starts with some other number n other than 1 (like 2 or 3, etc)? In other words, the first term is when n = some other number...

Homework Statement Prove or give a counterexample: If {an} and {an + bn} are convergent sequences, then {bn} is a convergent sequence. 2. The attempt at a solution Ok I couldn't think of any counterexamples, so I tried to prove it using delta epsilon definitions: |an - L| < E |an...

I get the impression that unlike solving derivatives and integrals, sequences and series do not have a lot of...should I say...find-the-equation-and-solve-your-way element -- sorry if that comes out wrong. Maybe it seems to be less "rote math" and because of this, I'm having a hard time trying...

Hi everyone. I feel like I'm really close to the answer on this one, but just out of reach :) I hope someone can give me some pointers Homework Statement Let A1 \supseteq A2 \supseteq A3 \supseteq \ldots be a sequence of compact, nonempty subsets of a metric space (X, d). Show that...

Hope all of you had a good July 4th (for those that celebrate it)! Anyways, I'm trying to figure out series and sequences. I'm using Stewarts' Single Variable Calculus: Early Transcendentals, 6th. ed. For instance, under Section 11.4, the Comparison Tests, I don't understand how one...

Homework Statement Define R^\infty_f = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \exists k_0 \text{ such that } t^{(k})=0 \; \forall k\geq k_0 \} Define l^\infty = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \sup_{k\geq 1} |...

Homework Statement A mortgage is taken out for £80,000. It is to be paid by annual instalments of 5000with the first payment being made at the end of the first year that the mortgage was taken out. Interest of 4% is then charged on any outstanding debt. Find the total time taken to pay off the...

Homework Statement Let (X,d) be a metric space with two sequences (x_n), (y_n) which converge to values of a,b respectively. Show that \lim_{n \to \infty} d(x_n,y_n) = d(a,b) Homework Equations (x_n) \rightarrow a \Leftrightarrow \forall \epsilon >0 \quad \exists n_0 \in...

In all the stuff I read, enhancer only binds activators while silencer sequences on DNA binds repressors. However, in my biology's slides.. my professor is saying that repressors can bind to both silencers and enhancers.. can anyone confirm this? This is very fishy as I have never read this...

Homework Statement Determine whether or not the sequences below are summable: {(-1)^n} {(-1)^n + (-1)^(n+1)} {(-1)^n} + {(-1)^(n+1)} Homework Equations The Attempt at a Solution Okay, I'm having some trouble thinking about these the right way. Since {(-1)^n}= -1, 1...

Sequences HELP! Homework Statement Show that the sequence Cn = [(-1)^n * 1/n!] Homework Equations The Attempt at a Solution This is an example in my book but I am not understanding it... It says to find 2 convergent sequences that can be related to the given sequence. 2 possibilities are...

Hey all! I wished someone tell me what I am doing wrong. The question asks to: Determine the sum of the following series I had : 16/7 as an answer. May I know what I am doing wrong?

Homework Statement (a) Show that \sum \frac 1n is not convergent by showing that the partial sums are not a Cauchy sequence (b) Show that \sum \frac 1{n^2} is convergent by showing that the partial sums form a Cauchy sequenceHomework Equations Given epsilon>0, a sequence is Cauchy if there...

Homework Statement Prove that the following sequence is Cauchy: a_n = [a_(n-1) + a_(n-2)]/2 (i.e. the average of the last two), where a_0 = x a_1 = y Homework Equations None The Attempt at a Solution I was trying to use the definition of Cauchy (i.e. |a_m - a_n| < e) by...

This is review from class the other day that I managed to miss because of illness and I was wondering if someone could explain how to go about solving these problems: #1 Let B = \left\{ \frac{(-1)^nn}{n+1}:n = 1,2,3,...\right\} Find the limit points of B Is B a closed set? Is B an open set...

Homework Statement Let u_n, v_n be a sequence of positive real number such that u_{n+1}\leq (1+v_n)u_n where \sum_{n=1}^{\infty}v_n converges. It is true or false if true please help me prove. if \frac{u_{n+1}}{u_n}\leq M then u_n\leq M for some M be a positive real number. whether...

Homework Statement In how many ways can we form a sequence of non-negative integers a_1,a_2,...,a_(k+1) such that the difference between the successive terms is 1 (any of them can be bigger) and a_1 =0. Homework Equations The Attempt at a Solution For k=1, there is only one...

The sequence {an} is defined by a1=1,an+1=\sqrt{1+an/2} ,n=1,2,3,4,... (a) a^2n-2<0 , (b)a^2a+1-a^2n>0. deduce that{an} converges and find its limits? please help me get the answer...

in general, how does one go about proving limits of recursive sequences using the definition? for example, how does one prove a_n+1 = (a_n)^2/5 => lim(a_n)=0? For me it's obvious but the TA insists that things like that require proving?

[SOLVED] Sequences in lp spaces... (Functional Analysis) Homework Statement Find a sequence which converges to zero but is not in any lp space where 1<=p<infinity. Homework Equations N/A The Attempt at a Solution I strongly suspect 1/ln(n+1) is a solution. Since ln(n+1) ->...

Homework Statement Let the sequence {a_n} defined by: a_n+1 = a_n/[sqrt(0.5a_n + 1) + 1] Prove that {a_n} converges to 0 Homework Equations The Attempt at a Solution I tried manipulating the equation but to no avail...

Can somebody please explain to me: How both series and sequences converge, and the various tests to find out. I've tried searching but it seems impossible to get any explanations as to why you do the specific test.

Homework Statement Give an example of two monotonic sequences whose sum is not monotonic Homework Equations nonoe The Attempt at a Solution Well, I'm thinking is you just used n and -n, would that be a valid attempt at the question, or is that just the lazy way out...

Prove that: ∀ n€N [(the) sum of an (infinite?) series (a1,+a2,...+,an)] (where a_{n}=\frac{n}{(n+1)!}) \sum \frac{n}{(n+1)!} (is equal to/gives/yields) = 1 - \frac{1}{(n+1)!} Prove that: ∀ n \in N \sum \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!} THX in advance

Homework Statement I need to find three mathematical series, for the following patterns 1) Sn= 1/n if n is odd, or 1 of n is even. 2) Sn= 0,1,0,.5,1,0,1/3,2/3,0,1/4... 3) Find a sequence (Sk) which is a subset of the natural numbers in which every positive integer appears infinitely...

Homework Statement Prove by an example that the sum or product of two non convergent sequences can be convergent Homework Equations There are none, they can be any sequences I guess The Attempt at a Solution I've tried a lot of possibilities. My first guess would be a series...

Dear all, I would highly appreciate if you could help me solving the following sequences: Question 1: 3,5,8,24,209,3591,? 33811 34308 35534 35200 35010 Question 2: 4,7,13,21,34,55,88,? 110 148 123 138 Question 3: 8,12,18,27,42,70,126,241,? 478 441 503 488 486...

Sequences, need help! Well there is this problem that i am struggling to proof, i think i am close but nope nothing yet. Well, the problem goes like this: We have two sequences \ {(a_n)} and ({b_n}), whith the feature that {a_n}<{b_n} for every n. Also, {a_2}={b_1}, {a_3}={b_2}... and so...

http://img215.imageshack.us/img215/8624/limitscz6.jpg That's the question and the answer. I don't get why it converges to -1/2 and how you can tell just by looking. Maybe I'm not understanding well enough but I understand how to find the limit, just I thought the limit to infinity showed...

Homework Statement how do show whether the following sequences are bounded? 1) {an}=sqrt(n)/1000 2) {an}=(-2n^2)/(4n^2 -1) 3) {an}=n/(2^n) 4) {an}=(ncos(npi))/2^n Homework Equations i have to show whether the sequences are bounded by a number but i don't know how to find that number...

Hello, I hope I have posted this in the correct place, if not, sorry. Okay, I have just started working through the material, but I am having some problems working out one of the examples, which is given below: Un+2 = 12Un+1 - 20Un (n = 0,1,2...) We are given U0 = 1, U1 = 2 and...

Hellou! I have a question regarding the square summable sequences: I should find an example of a closed set from the square summable sequences and show that the closed set does not have an element with a min norm! The professor mentioned an example: (1+1/1 0 0 0...) (0...

Hi, I'm trying to figure out my TI-89. So I want to estimate the 40th partial sum of this series: Sum(40) of (-1^(k+1))/k^4, starting at k=1. My major problem is that I want to use 'k's or 'n's, not 'x's. Is there a difference? I haven't asked my Calc teacher about this yet, but I know that...