Sequence Definition and 1000 Threads

Clearly, ##x_{n+1}>x_n \because x_n + \sqrt{x_n} > x_n## $$ \begin{align*} x_{n+1} &= x_n+ \sqrt{x_n} \\ &= x_1 + \sqrt{x_1} + \sqrt{x_2} + \cdots \sqrt{x_n} \\ &>n+1 \end{align*} $$ ##\because \sqrt{x_n}>\sqrt{x_1}=1## In fact, $$x_{n+1} > 1+ \sqrt{1} + \sqrt{2}+ \sqrt{3} + \cdots \sqrt{n}$$...

I attempted to solve it $$ x = \frac {1}{4x} + 1 $$ $$⇒ x^2 -x -\frac{1}{4} = 0 $$ $$⇒ x = \frac{1±\sqrt2}{2} $$ However, I don't know the next step for the proof. Do I need a closed-form of xn+1or do I just need to set the limit of xn and use inequality to solve it? If I have to use...

Summary:: x Let ## \{ a_{n} \} ## be a sequence. Prove: If for all ## N \in { \bf{N} } ## there exists ## n> N ## such that ## a_{n} \leq L ## , then there exists a subsequence ## \{ a_{n_{k}} \} ## such that ## a_{n_{k}} \leq L ## My attempt: Suppose that for all ## N \in {\bf{N}} ##...

Let $a,\,b,\,c$ be non-zero real numbers such that $(ab+bc+ca)^3=abc(a+b+c)^3$. Prove that $a,\,b,\,c$ are terms of a geometric sequence.

I saw the information of the text at different places and when i compare it is confusing. The text is In some other section If you see the red lines i have highlighted, in the first section the book refers that the B coil is behind the A coil, but in the 2nd section it refers that in the...

To find how much would be in the account after ten years, let the balance in the account at the start of year n be bn. Then b1=2000 I believe that this a compound interest problem. Common ratio r = 1.06 bn =2000*1.06^n−1 Thus, b10 =2000×1.06^9 = £3378.95791 The balance of the account at the...

1. When n=1, u1+1=3-1/3(u1) u2=3-1/3(3) u2=2 When n=2 u2+1=3-1/3(u2) u3=3-1/3(2) u3=7/3 When n=3 u3+1=3-1/3(u3) u4=3-1/3(7/3) u4=20/9 The common ratio is defiend by r=un+1/un, but this is different between the terms, i.e. u2/u1=2/3 whereas u3/u2=(7/3)/2=7/6 Have I made a mistake? 2. A...

b_{k} = b_{k - 1}/2 +b_{k-1} b_{0} = 1 What would be the sequence for this expression, I calculated it to be 1, 2/3, 2/5, 2/7 ... Is it right? My explicit formula is b_{n} = 2/n+2 What would be the explicit formula in your view and how can that formula be proved by mathematical...

Hey! 😊 Let $0<\alpha \in \mathbb{R}$ and $(f_n)_n$ be a sequence of functions defined on $[0, +\infty)$ by: \begin{equation*}f_n(x)=n^{\alpha}xe^{-nx}\end{equation*} - Show that $(f_n)$ converges pointwise on $[0,+\infty)$. For an integer $m>a$ we have that \begin{equation*}0 \leq...

I tried to use the ratio test, but I am stuck on finding the range of the limit. $$\because \left|x-1\right|<1.5=Radius$$ $$\therefore -0.5<x<2.5$$ $$\lim _{n \to \infty} \left| \frac{A_{n+1}(x-1)^{n+1}}{A_n(x-1)^n} \right|$$ $$\lim_{n \to \infty} \frac{A_{n+1} \left|x-1\right|}{A_n} <1$$...

I am trying to understand the math on this text: Why is not this sequence below right?: At first, +L, so the man inverts the wheel and he now has +2L So he give the wheel to assistent, who inverts the wheel, now wheel +L, the assistant give it to the man. The man + wheel have now +3L, the...

We have the limit of the sequence ##\frac{a^n}{n}## where ##a>1##. I know it is ##+\infty## and i can prove it by switching to the function ##\frac{a^x}{x}## and using L'Hopital. But how do i prove it using more basic calculus, without the knowledge of functions and derivatives and L'Hopital...

First I assume that $$(1+n^2)^\frac{1}{\ln n}=\exp {\ln (1+n^2)^\frac{1}{\ln n}} $$But, $${\ln (1+n^2)^\frac{1}{\ln n}}={\frac{\ln (1+n^2)}{\ln n}}$$ By L'Hopital Rule, I got $$\lim_{n\to\infty} {\frac{\ln (1+n^2)}{\ln n}}=\frac{\lim_{n\to\infty} (\frac{2n}{1+n^2})}{\lim_{n\to\infty} 1/n}$$...

Define a metric on ##\mathbb{R}[x]## for distinct polynomials ##f(x),g(x)## as ##d(f(x),g(x)) = \frac{1}{2^{n}}##, where ##n## is the largest positive integer such that ##x^{n}## divides ##f(x)-g(x)##. Equivalently, ##n## is the multiplicity of the root ##x=0## of ##f(x)-g(x)##. Set...

I think the initial assumptions would allow me to prove this without induction. Suppose ##(x_n)## is a real sequence that is bounded above. Define $$ y_n = \sup\{x_j | j \geq n\}.$$ Let ##n \in \mathbb{N}##. Then for all ##j \in \mathbb{N}## such that ##j \geq n + 1 > n## $$ x_{j} \leq y_n.$$...

I need a little help with Baby Rudin material regarding the convergence of a sequence of sets please. I wish to follow up on this thread with a definition of convergence of a sequence of sets from Baby Rudin (Principles of Mathematical Analysis, 3rd ed., Rudin) pgs. 304-305: (pg. 304)...

I need help only in section 3 I have some kind of solution but I'm not sure because it seems too short and too simple. We showed in section 1 that an> 0 per n. it Given that a_n + 1 <0 and a_n+1<\frac a_n a_1 In addition therefore a_1 <0 is warranted

##\textbf{Attempt at solution}##: If I can show that ##f## is integrable on ##[a,b]##, then for the second part I get : Let ##\frac{\varepsilon}{b-a} > 0##. By definition of uniform convergence, there exists ##N = N(\varepsilon) > 0## such that for all ##x \in [a,b]## we have ##\vert f(x) -...

Hi all; I have a very basic understanding of sequences and series and recently encountered a sequence which really has me confused: $$(\frac{1}{5}+(\frac{1}{5}+(\frac{1}{5}+(...)^2 )^2)^2)^2$$ What type of sequence would you call this? I couldn't even google it because I couldn't work out how to...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding the proof of Proposition 8.7 ...Proposition 8.7 and...

Hello! (Wave) Consider the sequence $a_n=\frac{8^n}{n!}$. It holds that $$a_{n+1}-a_n=\frac{8^{n+1}}{(n+1)!}-\frac{8^n}{n!}=\frac{8 \cdot 8^n}{(n+1) \cdot n!}-\frac{8^n}{n!}=\frac{8^n}{n!}\left( \frac{8}{n+1}-1\right)=\frac{8^n}{n!} \left( \frac{7-n}{n+1}\right)$$ Since the last term is...

Example done in class: The problem and my solution: My solution seems incorrect because if I try to plug in 0, I don't get the initial condition given in the problem. Does anyone see what I've done wrong along the way? Thanks.

I need some help with this task. My theory book only shows examples of how to solve sequences in the form : 𝑎𝑘 = A * 𝑎(𝑘−1) − B * 𝑎(𝑘−2). But I've no idea how to solve this task because of the alternating term. I've included the Answer (called "Svar") to the task.

Hello! I'm currently in the process of making a moving cosplay prop, and as a perfectionist, I'd like it to move exactly like the original. Kind of like in this animation: https://www.deviantart.com/ace-wong/art/RWBY-Qrow-s-Weapon-Animation-606779918 The problem I've stumbled upon, is how to...

Consider a function $f\in\mathcal{C}^2$ with Lipschitz continuous gradient (with constant $L$)- we also assume the function is lowerbounded and has at least one minimum. Let $\{x^k\}_k$ be the sequence generated by Gradient Descent algorithm with initial point $x^0$ and step-size $0<\alpha<2/L$...

Introduction: Making a Sequence ##T## based on “The Rule of Three” The primary means of generating the sequence ##T## is through the use of a function ##f##. In general, function ##f## is going to be a function that takes as input a three-member sequence of ordinal numbers (an ordered triplet)...

This problem arose in modeling camera focusing movement, such as a control system might do. It assumes a simple (thin) lens, rays close to the optical axis, and monochromatic light. While most camera lenses are not simple, this is a first approximation. Camera lenses project an image of a...

I was wondering how to deal with this step in Peskin and Schroeder: So first you make the delta fn. from the exponential as d(p - (- p')), then what do you do with the creation/annihilation operators that have a negative subscript? I don't have to go into position/momentum representation do I?..

So there are two parts of the question: a) does the sequence converge or diverge b) use nth term on the Series Now sybomlab calculator is saying to apply ratio test! a) b) So should I apply ratio test or is there any easy method? And what's the difference between these two questions and...

Is it possible for order to arrive out of disorder on a macro scale? Contrary to the 2nd law? Specifically, is it scientifically acceptable to believe that the random mutation of genetic material, which was itself produced by the random coupling of molecules has resulted, over the extended...

Let A ⊆ R, let f : A → R, and suppose that (a,∞) ⊆ A for some a ∈ R. Then the following statements are equivalent: i) limx→∞ f(x) = L ii) For every sequence (xn) in A ∩ (a,∞) such that lim(xn) = ∞, the sequence (f(xn)) converges to L. Not even sure how to begin this one, other than the fact...

So what I know about the Monotone Convergence Theorem is that it states that: if a sequence is bounded and monotone, then it is convergent. So all I have to show is that the sequence is bounded and monotone. My attempt at showing that it is bounded: The sequence can be expanded as: $$={\frac 1...

When viewing an HR diagram, the main sequence curve is apparent, and the general shape of it is obvious. However, in this truncated version, it's unclear to me exactly which stars should be considered main sequence. I've added a shaded grey area as what I think I should count as main sequence...

Hi. After some study, I came across symmetrical components. I found the attached schematic below on the internet, and will use it to explain my question. Question: Imagine that a single-line to Earth fault occur in one of the phases, i.e. resulting in an unbalanced system (we assume it was...

Consider the sequence $\{X_n\}_{n\geq 1}$ of independent random variables with law $N(0,\sigma^2)$. Define the sequence $Y_n= exp\bigg(a\sum_{i=1}^n X_i-n\sigma^2\bigg),n\geq 1$ for $a$ a real parameter and $Y_0=1.$ Now how to find the values of $a$ such that $\{Y_n\}_{n\geq 1}$ is martingale...

Let $x_{0}=1$ and $x_{n+1}=(-1)^{n}(\frac{\pi }{2}-\arctan(\frac{1}{x_{n}}))$ I have the following options to choose from: 1. $x_n$ is unbounded 2. $x_n$ is increasing and the limit of $x_n$ is $1$ 3. the limit of $x_n$ is $\pi/2$. 4. the limit of $x_n$ is $0$ My attempt: I used...

One can have a progression and it is called a Sequence. One can sum the terms in a sequence or progression, and this is called a Series. Why those terms like that; or why those two different terminologies? Was it decided just to pick a word Series so as to avoid the need to use Sum Of the...

This is no homework. I have come across a conjecture in a book called The art of the infinite:the pleasures of mathematics. I want to understand how to prove it. Homework Statement Consider a 3-rhythm starting with 2: ## 2, 5, 8, 11, 14, 17...## The each number in this sequenc has the form...

Suppose ##(y_n)_n## is a sequence in ##\mathbb{C}## with the following property: for each sequence ##(x_n)_n## in ##\mathbb{C}## for which the series ##\sum_n x_n## converges absolutely, also the series ##\sum_n \left(x_ny_n\right)## converges absolutely. Can you then conclude that ##(y_n)_n##...

I have the following sequence $(x_{n})$ , $x_{n}=1+\frac{1}{2^{2}}+...+\frac{1}{n^{2}}$ which has the limit $\frac{\pi ^{2}}{6}$.I need to calculate the limit of the sequence $(y_{n})$, $y_{n}=1+\frac{1}{3^{2}}+...+\frac{1}{(2n-1)^{2}}$ I don't know how to start.I think I need to solve the limit...

Homework Statement - Given a bounded sequence ##(y_n)_n## in ##\mathbb{C}##. Show that for every sequence ##(x_n)_n## in ##\mathbb{C}## for which the series ##\sum_n x_n## converges absolutely, that also the series ##\sum_n \left(x_ny_n\right)## converges absolutely. - Suppose ##(y_n)_n## is...

I have the following sequence: $x_{n}=(-1)^{n-1}\cdot \left(2+\frac{3}{n}\right),\forall n\in \mathbb{N^{*}}$ I need to choose from: A. $x_{n}$ is a monotonic sequence B. $x_{n}$ limit is $2$ C. $x_{n}$ minimum is $-\frac{7}{2}$ and the maximum is $5$ D. $x_{n}$ minimum is $-2$ and the maximum...

I have the following sequence $(a_{n})$, $a_{1}=1$ $$a_{n+1}=\begin{cases} a_{n}+\frac{1}{2} & \text{ if } n \ is \ even \\ \frac{a_{n}}{3} & \text{ if } n \ is \ odd \end{cases}$$ I need to find $$\lim_{n\rightarrow \infty }a_{2n+1}$$ I tried something but I didn't get too far.I rewrite the...

I have the sequence from the picture and I have to demonstrate that this sequence has a limit. I always get stuck at this kind of exercises.How to approach an exercise like this?

Homework Statement Hi I am looking at this proof that , if on an open connected set, U,there exists a convergent sequence of on this open set, and f(z_n) is zero for any such n, for a holomorphic function, then f(z) is identically zero everywhere. ##f: u \to C##Please see attachment...

<Moderator's note: Moved from a technical forum and thus no template.> Suppose that all stars in this galaxy were born in a single major-merger burst event about 10 Gyr ago. From this original burst, I want to compute the fraction of stellar mass still surviving as stars in the main sequence ...

Homework Statement Determine whether or not the following sequences of real valued functions are Cauchy in L^{1}[0,1]: (a) f_{n}(x) = \begin{cases} \frac{1}{\sqrt{x}} & , \frac{1}{n+1}\leq x \leq 1 \\ 0 & , \text{ otherwise } \end{cases} (b) f_{n}(x) = \begin{cases} \frac{1}{x} & ...

Hi, I have this equation about Rossler system, describe as Eq. 1. Given that the chaotic behavior of the system for parameter values a=b=0.2 and c=5.7. How can I calculate the chaotic sequence for this equation below. The equation also referred from...

Hello! (Wave) Let $(a_n), (b_n), (c_n)$ sequences such that $(a_n), (c_n)$ are bounded and $a_n \leq b_n \leq c_n$ for each $n=1,2, \dots$ I want to show that $(b_n)$ has a convergent subsequence. I have thought the following: Since $(a_n), (c_n)$ are bounded, $\exists m_1, m_2 \in...