Sequence Definition and 1000 Threads

Homework Statement suppose that 0≤xm+n≤xm+xn for all m,n∈ℕ, prove that the limit of xn/n exists when n tends to infinity. Homework EquationsThe Attempt at a Solution I get that xn is bounded by zero and x1. And I guess that xn is monotonous but i find it hard to prove. Or maybe there is...

Hello! (Wave) Let $0< \theta<1$ and a sequence $(a_n)$ for which it holds that $|a_{n+2}-a_{n+1}| \leq \theta |a_{n+1}-a_n|, n=1,2, \dots$. Could you give me a hint how we could show that $(a_n)$ converges? :confused:

I would like to ask a question about MRI Spin echo sequence. after first 90 degree RF pulse, the free induction decay occurs. And 180degree refocusing pulse is given again. And echo signal is obtained at TE. My question is that is the the signal highest when the protons are back again in...

Say that we are asked to prove, using the definition of limits, that the sequence ##\frac{4n^2+3}{n^2+n+2}## tends to ##4## as ##n## tends to infinity. The following is a screenshot of the solution I found in a YouTube video: (Note that in the definition above, "g" denotes the limit - in this...

Hello! (Wave) I want to show that for each $n \geq 1$ it holds that $2^n L_n \equiv 2 \pmod{10}$. $L_n$ is the Lucas sequence. According to my notes, $$L_n=\left( \frac{1+\sqrt{5}}{2}\right)^n+\left( \frac{1-\sqrt{5}}{2}\right)^n$$ and $$L_n=F_{n-1}+F_{n+1},$$ where $F_n$ is the $n$-th...

Consider a spring balance with no initial deflection. Let an object of mass 'm' be attached to it. We allow the spring to come into equlibrium, and 'd' is the deflection at this eqb position. We add another object of mass 'M', while m is also present, so that the final position is x, and hence...

Homework Statement Modify the function so that instead of returning all n numbers, it only returns the nth number. Homework EquationsThe Attempt at a Solution I'm not sure how to return only the nth number of the fibonacci sequence. Please help.

Homework Statement This is a translation so sorry in advance if there are funky words in here[/B] f: ℝ→ℝ a function 2 time differentiable on ℝ. The second derivative f'' is bounded on ℝ. Show that the sequence on functions $$ n[f(x + 1/n) - f(x)] $$ converges uniformly on f'(x) on ℝ...

Homework Statement Let ##X \subset \mathbb{C}##, and let ##f_n : X \rightarrow \mathbb{C}## be a sequence of functions. Show if ##f_n## is uniformly Cauchy, then ##f_n## converges uniformly to some ##f: X \rightarrow \mathbb{C}##. Homework Equations Uniform convergence: for all ##\varepsilon >...

Hello! (Wave) I want to check as for the convergence the sequence $(a^n b^{n^2})$ for all the possible values that $a,b$ take. I have thought the following: We have that $\lim_{n \to +\infty} a^n=+\infty$ if $a>1$, $\lim_{n \to +\infty} a^n=0$ if $-1<a<1$, right? What happens for $a<-1$ ...

Hello! (Wave) Let $(a_n)$ be a sequence of real numbers such that $a_n \to a$ for some $a \in \mathbb{R}$. I want to show that $\frac{a_1+a_2+\dots+a_n}{n} \to a$. We have the following: Let $\epsilon>0$. Since $a_n \to a$, there is some positive integer $N$ such that if $n \geq N$, then...

Let x denote the position of a particle on the number line. From x, it can move to either the point a-a2+ax or to the point x-ax-a+a2 for some fixed 0<a<1. Suppose the particle starts at the origin. Prove that any open interval that is a subset of the interval (a-1,a) contains a point that the...

Homework Statement With ##a_1\in\mathbb{N}## given, define ##\displaystyle {\{a_n\}_{n=1}^\infty}\subset\mathbb{R}## by ##\displaystyle {a_{n+1}:=\frac{1+a_n^2}{2}}##, for all ##n\in\mathbb{N}##.Homework EquationsThe Attempt at a Solution We claim that with ##a_1 \in \mathbb{N}##, the sequence...

Homework Statement Let ##x,y## be positive numbers. Let ##a_0 = y## and let ##a_n = \frac{(x/a_{n-1})+a_{n-1}}{2}##. Prove that ##(a_n)## is a decreasing sequence with limit ##\sqrt{x}##. Homework EquationsThe Attempt at a Solution I'm confused about the initial condition being an arbitrary...

Homework Statement Prove that every convergent sequence has a monotone subsequence. Homework EquationsThe Attempt at a Solution Define ##L## to be the limit of ##(a_n)##. Then every ##\epsilon##-ball about L contains infinitely many points. Note that ##(L, \infty)## or ##(-\infty, L)## (or...

Homework Statement [/B] Let ##S\subset \mathbb{R}## be nonempty and bounded above. Show that there must exist a sequence ##\{a_n\}_{n=1}^\infty\subseteq S## such that ##\lim_{n\to\infty}a_n=\sup(S)##. Homework EquationsThe Attempt at a Solution Here is my idea. Let ##\epsilon >0##. Then there...

Hi there. I am working with a problem where a sequence of numbers arises. This sequence reads: ##\{0,1,3,5,10,15,21,28\}## as far as I have worked it. I am trying to figure out the underlying relation that gives this sequence. These are related to indexes in a matrix, and I am trying to...

I have a Dover edition of Louis Brand's Advanced Calculus: An Introduction to Classical Analysis. I really like this book, but find his proof of limit laws for sequences questionable. He first proves the sum of null sequences is null and that the product of a bounded sequence with a null...

Homework Statement The first three terms of a GP are X,X+2,X+3. The value of X and the fifth term is.[/B] (a)-4,1/4 (b)4,1/4 (c)2,1/4 (d)-2,-1/4 Homework EquationsThe Attempt at a Solution (x+2/x)=(x+3)/(x+2) (x+2)2=x2+3x x2+2x+4=x2+3x x=4 so i think r=(x+2)/x putting x=4 r=3/2 also...

Homework Statement Fill in the dots: 83 80 84 83 88 95 ... Pick one of the following answers: 95 91 83 87 Homework Equations The Attempt at a Solution 84-83 = 1 88-84= 4 88 + 7 = 95 ?

In several places (e.g., page 12 of http://www.cs.williams.edu/~bailey/06le.pdf), I have come across the aperiodic intervals in a one-dimensional Penrose tiling as "musical sequences". I do not see the connection between aperiodicity and music. The history of a fruitless but amusing search: (a)...

Homework Statement [/B] I am reviewing an example on the basics of the genetic code; this example is listed at the bottom of the following webpage: https://www.atdbio.com/content/14/Transcription-Translation-and-Replication.I have produced the example below and added Roman numbers to better...

Given S is a submanifold of M such that every smooth function on S can be extended to a smooth function to a neighborhood W of S in M. I want to show that S is embedded submanifold. My attempt: Suppose S is not embedded. Then there is a point p that is not contained in any slice chart. Since a...

Let ##M## be a left R-module and ##f:M \to M## an R-endomorphism. Consider this infinite descending sequence of submodules of ##M## ##M \supseteq f(M) \supseteq f^2(M) \supseteq f^3(M) \supseteq \cdots (1)## Can anybody show that the sequence (1) is strictly descending if ##f## is injective...

Homework Statement in title Homework Equations n = 2,3,4... The Attempt at a Solution n!/(n-2)! = n!/(n!(n-2)) = 1/(n-2) lim n->∞ = 1/∞ = 0 so sequence converges Incorrect

Homework Statement ##a_1## = 3 ##a_{n+1}## = ##\frac{2}{3} a_n + \frac{1}{4} ## Homework Equations [/B]The Attempt at a Solution Sequence i got : ## a_1, a_2, a_3, a_4, a_5 ## ## 3, \frac{9}{4}, \frac{7}{4}, \frac{17}{12}, \frac{43}{36} ## I tried to find the formula of ##a_n## ##a_n =...

Suppose I have the sequence ##a_n = 2^{(-1)^n}##. So ##\displaystyle (a_n) = (\frac{1}{2},2,\frac{1}{2},2,\frac{1}{2},2,\frac{1}{2},2,...)##. Clearly, this sequence has two subsequential limits, ##\displaystyle \{\frac{1}{2},2 \}##. This clear from observation, but I'm not sure how I can be sure...

Homework Statement http://sites.math.rutgers.edu/~ds965/temp.pdf (NUMBER 2)[/B]Homework Equations I do not understand the alternating part for the second problem and the recursive part for the first problem.The Attempt at a Solution The first answer I got was first by writing out the...

Given a function in ##f \in L_2(\mathbb{R})-\{0\}## which is non-negative almost everywhere. Then ##w-lim_{n \to \infty} f_n = 0## with ##f_n(x):=f(x-n)##. Why? ##f\in L_2(\mathbb{R})## means ##f## is Lebesgue square integrable, i.e. ##\int_\mathbb{R} |f(x)|^2 \,dx< \infty ##. Weak convergence...

Homework Statement Let ##f## be a real-valued function with ##\operatorname{dom}(f) \subset \mathbb{R}##. Prove ##f## is continuous at ##x_0## if and only if, for every monotonic sequence ##(x_n)## in ##\operatorname{dom}(f)## converging to ##x_0##, we have ##\lim f(x_n) = f(x_0)##. Hint: Don't...

A geometric sequence has an initial value of 25 and a common ratio of 1.8. Write a function to represent this sequence . Find the 23rd term. My Effort: The needed function is a_n = a_1•r^(n-1), n is the 23rd term, r is the common ratio and a_1 is the initial value. a_23 = 25•(1.8)^(23 - 1)...

Homework Statement Prove that ##\displaystyle t_{n+1} = (1 - \frac{1}{4n^2}) t_n## where ##t_1=1## converges. Homework EquationsThe Attempt at a Solution First, we must prove that the sequence is bounded below. We will prove that it is bounded below by 0. ##t_1 = 1 \ge 0##, so the base case...

I have the following sequence: ##s_1 = 5## and ##\displaystyle s_n = \frac{s_{n-1}^2+5}{2 s_{n-1}}##. To prove that the sequence converges, my textbook proves that the following is true all ##n##: ##\sqrt{5} < s_{n+1} < s_n \le 5##. I know to prove that this recursively defined sequence...

Dear Everyone, Here is the sequence: Let $S\subset\Bbb{R}$ and ${x}_{n}\in S$ and $S\ne\emptyset$ . ${x}_{n-1}<{x}_{n}\le\sup S$ for all $n\ge2$. Prove the sequence is monotone increasing. I need help proving it; I do not know where to start? Thanks Carter

Homework Statement Given that ##t_1 = 1## and ##\displaystyle t_{n+1} = \frac{t_n^2 + 2}{2t_n}## for ##n \ge 1##. Show that ##t_n > 0## for all ##n##. Homework EquationsThe Attempt at a Solution Intuitively this is obvious. Since ##t_1## is positive, so is ##t_2##, and so on. But I am having...

Hey PF! This isn't for homework, just me messing around with some thoughts in caluclating various homology groups. So suppose we have ##p \in S^n## and suppose that ##X## is a Polyhedra. I want to show that ##H_q(X \times S^n, X \times p) \cong H_{q-n}(X)## I was given the hint to start out...

I need help with this problem... By experimenting with numerous examples in search of a pattern, determine a simple formula for (Fn+1)2−(Fn−1)2; that is, a formula for the difference of the squares of two Fibonacci numbers.What does this question want? What is it asking for?

Homework Statement Prove rigorously that ##\displaystyle \lim \frac{n}{n^2 + 1} = 0##. Homework Equations A sequence ##(s_n)## converges to ##s## if ##\forall \epsilon > 0 \exists N \in \mathbb{N} \forall n \in \mathbb{N} (n> N \implies |s_n - s| < \epsilon)## The Attempt at a Solution Let...

Dear Every one, In my book, Basic Analysis by Jiri Lebel, the exercise states "show that the sequence $\left\{(n+1)/n\right\}$ is monotone, bounded, and use the monotone convergence theorem to find the limit" My Work: The Proof: Bound The sequence is bounded by 0. $\left|{(n+1)/n}\right|...

Homework Statement "Let ##\{a_n\}_{n=1}^\infty## be a bounded, non-monotonic sequence of real numbers. Prove that it contains a convergent subsequence." Homework Equations Monotone: "A sequence ##\{\alpha_n\}_{n=1}^\infty## is monotone if it is increasing or decreasing. In other words, if a...

Homework Statement [/B] The proposition that I intend to prove is the following. (From Terence Tao "Analysis I" 3rd ed., Proposition 6.1.7, p. 128). ##Proposition##. Let ##(a_n)^\infty_{n=m}## be a real sequence starting at some integer index m, and let ##l\neq l'## be two distinct real...

Homework Statement I think two diagrams are wrong here. I've marked in red.[/B] Homework Equations Do you also think the 2 diagrams are wrong? I think for 1st red circle, at delta-- the a1 switch should be open whereas for second red circle at star grounded-- the a1 switch should be closed...

Homework Statement It's a solved problem but I don't understand why is there no 30 degree lag from line voltage to lead voltage. Homework Equations Phase voltage = Line Voltage / 1.732 and there is 30 degree lag So shouldn't Ir be at angle 150 degrees.The Attempt at a Solution In Y line and...

I wonder if anyone can please help / point me to some info on how to solve this problem. I posted the same question on another website, and so far there is no conclusive answer. I have some pharmacokinetic data for a molecule that was administered in rat, first IV (intra-venously), then PO...

Hi, I am using mathlib as such: text...$\mathlib{L}^2[a,b]$...text though, I get the error: <recently read> \mathlib l.235 ...d form the complex vector space $\mathlib {L}^2[a,b]$ which satisfie... The control sequence at the end of the top line of your error message was never \def'ed. If you...

This isn't homework, it's a proof left to the reader as I self study Munkre's 'Elements of Algebraic Topology' Prove that if the sequence ##A_1 --> A_2 --> A_3 --> A_4 --> A_5## is exact Then so is the induced sequence: ##0 --> cok(a_1) --> A_3 --> ker(a_4) --> 0## where ##a_1## and ##a_4##...

I have in the back of my head the statement that for every finite sequence of positive integers there exists a pattern (i.e., a generating formula). While this sounds reasonable, I am not sure whether it is true, and if it is true, what the source for this statement is, and how the correct...

Homework Statement For some background, this is an advanced calculus 1 course. This was an assignment from a quiz back early in the semester. Any hints or a similar problem to guide me through this is greatly appreciated! Here is the problem: Find a convergent subsequence of the sequence...

Hey suppose I have sequence An limAn,n→∞ = ∞ Is it possible to find a sequence which makes: lim (An/An+1) ,n →∞ = ∞? I tried to search a sequence like that and could not find, but I don't know how to prove that this is can not be happening. could you help please?

In the textbook I have (its a textbook for calculus from my undergrad studies, written by Greek authors) some times it uses the lemma that "for any irrational number there exists a sequence of rational numbers that converges to it", and it doesn't have a proof for it, just saying that it is a...