Sequence Definition and 1000 Threads

Homework Statement . Let ##(X_n,d_n)_{n \in \mathbb N}## be a sequence of metric spaces. Consider the product space ##X=\prod_{n \in \mathbb N} X_n## with the distance ##d((x_n)_{n \in \mathbb N},(y_n)_{n \in \mathbb N})=\sum_{n \in \mathbb N} \dfrac{d_n(x_n,y_n)}{n^2[1+d_n(x_n,y_n)]}##...

Homework Statement Consider (R,C). Prove that a sequence converges in this topological space iff it is bounded below define ##C = ## ##\left \{ (a,\infty)|a\in R \right \} \bigcup \left \{ \oslash , R \right \}## Homework Equations The Attempt at a Solution So I am not very...

Homework Statement . Let ##(X,d)## be a metric space and let ##D \subset X## a dense subset of ##X##. Suppose that given ##\{x_n\}_{n \in \mathbb N} \subset X## there is ##x \in X## such that ##\lim_{n \to \infty}d(x_n,s)=d(x,s)## for every ##s \in D##. Prove that ##\lim_{n \to \infty} x_n=x##...

Homework Statement . Prove that ##\{x_n\}_{n \in \mathbb N} \subset \mathbb R## doesn't have any convergent subsequence iff ##lim_{n \to \infty} |x_n|=+\infty##. The attempt at a solution. I think I could correctly prove the implication ##lim_{n \to \infty} |x_n|=+\infty \implies## it...

I'm trying to do 3 questions, each one a bit more complex than the previous, but all have the same ideas. ( 2) has 1 more term than 1, 3) is with imaginary numbers) Could someone please guide me on how to do them? Am I trying to substitute things into each other? Suppose that the sequence x0...

What is the next logical number in the sequence: 1, 1, 3, 6, 18, ??

Homework Statement What is the Fourier transform of a single short pulse and of a sequence of pulses? The Attempt at a Solution In class we haven't dealt with the mathematics of a Fourier transform, however my professor has simple stated that a Fourier transform is simply a equation...

Homework Statement . Let ##\{I_n\}_{n \in \mathbb N}## be a sequence of closed nested intervals and for each ##n \in \mathbb N## let ##\alpha_n## be the length of ##I_n##. Prove that ##lim_{n \to \infty}\alpha_n## exists and prove that if ##L=lim_{n \to \infty}\alpha_n>0##, then ##\bigcap_{n...

Hi ! :) Let x_{n} a sequence of positive numbers.How could I show that it has a decreasing subsequence that converges to 0,knowing that inf{ x_{n} ,n ε N} =0??

Real Analysis, L∞(E) Norm as the limit of a sequence. || f ||_{\infty} is the lesser real number M such that | \{ x \in E / |f(x)| > M \} | = 0 ( | \cdot | used with sets is the Lebesgue measure). Definition: For every 1 \leq p < \infty and for every E such that 0 < | E | < \infty we...

Let $k_1,k_2,\cdots$ be a sequence defined by $k_1=1$ and for $n \ge 1$, $k_{n+1}=\sqrt{k_n^2-2k_n+3}+1$. Find $k_{513}$.

Homework Statement {U_0 = 9, U_1 = -3} U_(n+2) = -(5/4) U_(n+1) + (3/8) U_(n) Homework Equations The Attempt at a Solution First step was to attempt to find the common difference by trying to find the 3rd term: U_(2) = -(5/4) u_(1) + 3/8 U_(0) = -(57/8) This does not...

Homework Statement . Given ##f_n=\frac {x} {1+x^2}-\frac {(1+x^2)x} {1+(n+1)^2x^2}## , prove that ##\{f_n\}_{n \in \mathbb N}## converges pointwise and uniformly to a continuous function on the interval ##[0,1]## The attempt at a solution. It's easy to prove that this sequence tends to...

In arithmetic sequence we know that a_1+a_3 = 6 and 3^{a_1+a_2}=243 a) Find the initial term of the sequence b) Calculate,how much members of the sequence we have to add (a_1+a_2+...a_n) that we get the result 243? Have no idea where to start :confused:

Homework Statement If ##x_1 < x_2## are arbitrary real numbers and ##x_n=\frac{1}{2}(x_{n-2}+x_{n-1})## for## n > 2##, show that ##(x_n)## is convergent.Homework Equations Definition of Contractive Sequence: We say that a sequence ##X=(x_n)## of real numbers is contractive if there exists a...

Homework Statement A sequence of numbers ##x_n## is determined by the equality ##x_n=\frac{x_{n-1}+x_{n-2}}{2}## and the values of ##x_0## and ##x_1##. Compute ##x_n## in terms of ##x_0, x_1## and ##n##. Also prove that $$\lim_{n \rightarrow \infty} x_n=\frac{x_0+2x_1}{3}$$.Homework Equations...

Let $s_n$ be a sequence of real numbers and define $E$ to be the set of all subsequential limits of $s_n$ in the real extended line. Then we define the following \lim \text{sup } s_n = \text{sup } E For some reason I don't quite understand the above formula , do we need to prove it ? It would...

Hi everyone, :) Can somebody give me a hint to solve this problem. :) Problem: Let \(f\) be a function defined on \([a,\,b]\) with continuous second order derivative. Let \(x_0\in (a,\,b)\) satisfy \(f(x_0)=0\) but \(f'(x_0)\neq 0\). Prove that, there is a neighbourhood of \(x_0\), say...

Homework Statement a) (1+i)-n as n→∞ b) n/(1+i)n as n→∞ Homework Equations The Attempt at a Solution My answers were divergent for both question because (1+i)n=sqrt(2)*en*pi*i/4, so when n→∞, the limit is varying on the circle with radius sqrt(2). But the solution said both of...

Homework Statement Could you guys please verify this proof of mine? I want to show that a limit with particular property does not go to 0. It is part of the proof that when a sequence have an ever increasing term then the limit of the sequence is not 0. The Attempt at a Solution The...

Homework Statement Find a, sucht that: \lim_{ x \to \infty }( a \sqrt{n+2} - \sqrt{n+1} ) ) = \infty(a+1) Now, I want this sequence to have the limit 0. The first impule is to say that a+1 = 0 and hence a = -1. But if I do this I get \infty 0 which can't be determined. The paradox is...

Homework Statement The following sequence comes from the recursion formula for Newton's Method. x0= 1 , xn+1=xn-(tanxn-1)/sec2xn Show if the sequence converges or diverge. Homework Equations The Attempt at a Solution I don't really know where to start on this problem, I have tried to use some...

Homework Statement Prove that if ##0<c<1## then llim##c^{\frac{1}{n}}=1## using the monotone convergence theorem. Homework Equations The Attempt at a Solution I let ##c_n=c^{\frac{1}{n}}## and it follows since ##0<c<1 \implies 0<c^{\frac{1}{n}}<1## Thus ##c_n## is bounded above...

Homework Statement Let f_{n}(x)=\frac{x}{1+x^n} for x \in [0,∞) and n \in N. Find the pointwise limit f of this sequence on the given interval and show that (f_{n}) does not uniformly converge to f on the given interval. Homework Equations The Attempt at a Solution I found that the pointwise...

show if the sequence ## x_n## is bounded and ## y_n \rightarrow + \infty ## then ## x_n + y_n \rightarrow + \infty ## my attempt if ## x_n ## is bounded then ## P \leq x_n \leq Q ## for some ## P,Q \in \mathbb{R} ## if ## y_n \rightarrow + \infty ## then ## \forall M>0 ## ## \exists N \in...

Please help! I need to be able to find the number of total toothpicks used in any given figure in this sequence, in addition to total squares in any figure, using the figure number. The only way I can figure this out is by listing them all out one by one, which is TERRIBLE :( Thank you so much...

HelloI want to prove the following. Let \(X\) and \(Y\) be two sequences,and \(XY\) converges. Then prove that \(X_mY\) also converges,where \[ X_m = \mbox{ m-tail of X } = (x_{m+n}\;:\; n\in \mathbb{N}) \] Here is my proof. let \(\lim\;(XY) = a \) . Then we have \[ \forall \varepsilon >0\...

I have one more problem,for the given sequence i have to find the exact upper and lower limit,and to argument them. i have been missing on this lesson,so please help me,i don't know how to do it. So the sequence is: an = \frac{2n+3}{n}

Im new to limits,so please can anybody help me to solve those? I have to find a limits for given sequences. Detailed instruction how to solve this would be great. Thank you! 1. \lim _{n \to \infty} \frac{2}{3} + \frac{3}{2n^2} 2. \lim _{n \to \infty} \frac{5n^3+6n-3}{7n-3n^3+2} 3. \lim _{n...

question: Suppose ## S \subset \mathbb{R} ## is a nonempty subset of the real numbers that is bounded below. Show that there exists a sequence ## <x_n> ## such that ## x_n \in S ## for all n and ## \lim (x_n) = inf(S) ## attempt: consider an element ## x \in S ## suppose ## x \geq inf(S) +...

Can somebody help me with this problem? I have to proof if 41/81 is a part of attached sequence. Step by step guide would be useful. Thank you! http://img.tapatalk.com/d/13/10/18/2unyje8y.jpg

Homework Statement Question from my professor: "Consider the sequence {a(n)} (from n=1 to ∞) defined inductively by a(1) = 0, and a(n+1) = √(a(n) + 2) for n ≥ 1. Prove that {a(n)} (from n=1 to ∞) is increasing". Here's the first part of the answer from my professor: "Consider...

Homework Statement find the limit z_n = [(1+i)/sqrt(3)]^n as n -> ∞. Homework Equations [b]3. The Attempt at a Solution Apparently the limit is zero (via back of the book), but I have no clue how they got that answer. (1 + i)^n seems to be unbounded, thus i do not see how z_n can go to...

Homework Statement a real sequence (x_{n}) is defined as follows: we take the elements in order (starting from x0) to be 0, 1 , 0 , 1/10 , 2/10 ,... , 9/10, 1 0 , 1/100 ,2/100 ,..., 99/100 , 1 , 0 , 1/1000,... So we take p for p = 0, 1, then p/10 for p = 0; ... 10, then p=100 for p =...

Homework Statement 2.11. Determine (explicitly) a convergent subsequence of the sequence in R2 given for n = 1; 2; : : : by xn =(e^{n}sin(n\pi/7),((4n+3/3n+4)cos(n\pi/3)) I know that the Bolzano-weierstrass theorem says that every bounded sequence has a convergent subsequence. I...

HelloI want to prove the following. \[ \lim_{n\rightarrow \infty}\left((2n)^{1/n}\right) = 1 \] where \( n \in \mathbb{N} \). Now since we have nth root of a positive number, I used theorem on the existence of nth root to argue that \( (2n)^{1/n} > 0 \). Next I tried to prove that \( (2n)^{1/n}...

one of example of cauchy sequence show that = 1/n - 1/(n+k) and In the above we have used the inequality 1/(n+m)^2 <= ( 1/(n+m-1) - 1/(n+m) ) => i don't under stand where this come from and what is inequality? can you give other example?

I have a problem and honestly have no idea even where to start. I've been staring at it and thinking about it for over 24 hours... Let \(u_n\) denote the \(n^{th}\) Fibonacci number. Without using the Binet formula for \(u_n\), prove the following for all natural numbers \(m\) and \(n\) with...

I would like to know if there is a general formula, and if so, what it is, for finding the $limsup$ and $liminf$ of a sequence of sets $A_n$ as $n\rightarrow \infty$. I know the following examples: **(1)** for $A_n=(0,a_n], (a_1,a_2)=(10,200)$, $a_n=1+1/n$ for $n$ odd and $a_n=5-1/n$ for $n$...

Homework Statement Q1 Are the following sequences divergent or convergent as n tends to infinity. A: \frac{5n+2}{n-1} B: tan^{-1}(n)Homework Equations The Attempt at a Solution Really not sure how to show this mathematically, or even if what I have done is correct. Part A...

Homework Statement ... if fj are holomorphic on an open set U and fj \stackrel{uniformly}{\rightarrow} f on compact subsets of U then δ/δz(fj) \stackrel{uniformly}{\rightarrow} δ/δz(f) on compact subsets of U. Give an example to show that if the word "holomorphic" is replaced by "infinitely...

Hey, I would like to know if there are any steps to follow to get a formula or a rule for a sequence, or does it depend on logic and practicing ?? Thanks :)

1. Homework Statement Provide a simple formula or rule that generates the terms of an integer sequence that begins with : * 2,4,16,256,65536,... 3. The Attempt at a Solution I have tried a lot to solve it but i ended up with nothing,although i know that finding a term in the...

I have to examine whether this sequence Xn = ln(n^2+1) - ln(n) converges or diverges. My attempt at a solution: Xn = ln(n^2+1) - ln(n) = ln((n^2+1)/n) = ln(n+1/n) Xn → ∞ when n → ∞ So the sequence diverges. Can someone look at this and see whether the procedure...

It can be shown that when $5n+1$ and $7n+1$ (where $n\in\mathbb{N}$) are both perfect squares, then $n$ is divisible by $24$. Find a method for generating all such $n$.

Hello. I am reading an introduction to induction example, and I am having the hardest time trying to determine what exactly happened in the proof. Can somebody please help? How can ##3^{k-1}## + ##3^{k-2}## + ##3^{k-3}## all of a sudden become ##3^{k-1}##+##3^{k-1}##+##3^{k-1}## and how can be...

The fibonacci sequence can be defined as $${F_n} = {F_{n - 1}} + {F_{n - 2}}$$ and specifying the initial conditions as $$\eqalign{ & {F_1} = 1 \cr & {F_2} = 1 \cr} $$  Also there exists a general formula for the fibonacci which is given by $${F_n} = {{{\varphi ^n} + {\psi ^n}} \over...

Let Fn denote the Fibonacci sequence. un is the sequence given by: un= Fn+1/Fn. Show that mod(un - \phi) \leq\frac{1}{\phi}mod(un-1-\phi) and therefore mod(un - \phi) \leq \frac{1}{\phin-1}[/itex]mod(u1-\phi) and then conclude un converges to \phi I have tried with the identity \phi = 1+...

Homework Statement If ##X=(x_n)## is a positive sequence which converges to ##x##, then ##(\sqrt {x_n})## converges to ##\sqrt x.## 2. The attempt at a solution I was given a hint: ##\sqrt x_n -\sqrt x = \frac{x_n-x}{\sqrt x_n +\sqrt x}.## How can I obtain that hint if it were never...