Sequence Definition and 1000 Threads

I found the following convergent sequence for the natural logarithm online: \lim_{a\rightarrow\infty}a x^{1/a}-a=\ln(x) Does anybody know where this sequence first appeared, or if it has a name?

Let the sequence $\left\{x_n\right\}$ of integers (modulo $11$) be defined by the recurrence relation: $x_{n+3} \equiv \frac{1}{3}(x_{n+2}+x_{n+1}+x_n)$ (mod $11$), for $n=1,2,..$ Show, that every such sequence $\left\{x_n\right\}$ is either constant or periodic with period $10$.

Homework Statement Determine if the degree sequence 3,3,2,2,2,2 is graphic. Homework Equations Havel-Hakimi The Attempt at a Solution [/B] Check to see if the sum is even: 3+3+2+2+2+2 = 16It is even, therefore apply Havel-Hakimi 3,3,2,2,2,2 -> remove the 3 and subtract the next 3 by 1...

Homework Statement Prove that \lim \frac{n+100}{n^{2}+1} = 0 Homework Equations (x_{n}) converges to L if \forall \hspace{0.2cm} \epsilon > 0 \hspace{0.2cm} \exists \hspace{0.2cm} N\in \mathbb{N} \hspace{0.2cm} \text{such that} \hspace{0.2cm} \forall n\geq N \hspace{0.2cm} , |x_{n}-L|<...

I needed help figuring out the formula for a sequence with these conditions: 1. The first term is 1. 2. All terms are positive integers. 3. 3 times each term (3*a_n) must be greater than the sum of that term and all the terms before it, but no more than 3 greater. Operators like ceil() and...

So I have this problem I'm stuck on wrapping my head around a particular problem "In the sequence a{n}, let a{0}=2. If a{n+1} = 3 a{n} −1, then what is the value of a3?" I understand it's following the pattern of each term, and that with Arithmetic sequence a{n-1} means you would use the a{n}...

Prove that the sequence :$$(-1)^n$$ diverges by using the ε-definition of the limit of a sequence

Homework Statement Given that ##\{x_n\}## is a bounded, divergent sequence of real numbers, which of the following must be true? (A) ##(x_n)## contains infinitely many convergent subsequences (B) ##(x_n)## contains convergent subsequences with different limits (C) The sequence whose...

##\displaystyle f_{n} = ((\sum_{k= \lfloor{\frac{n}{2}}\rfloor}^{n-1}f_{k}) mod 1) + 0.1## ##\displaystyle f_{1} = 0## I really would like to know where to begin for finding a formula for the nth term, I wrote out a bunch of the terms and couldn't really eyeball a pattern of any sort. I noticed...

Find $a_{2017}$, if $a_1 = 1$, and $$\frac{a_n}{n+1}=\frac{\sum_{i=1}^{n-1}a_i}{n-1}.$$

Hello, would like to derive a length of list of random numbers in which I may find some special sequence of few numbers with some probability. For clearness I give an example: I have two generator of (pseudo) random numbers with same range of numbers, let's say (1-k). First generator give a...

Is it possible to use Axiom of Choice to prove that there would exist a sequence (A_n)_{n=1,2,3,\ldots} with the properties: A_n\subset\mathbb{R} for all n=1,2,3,\ldots, A_1\subset A_2\subset A_3\subset\cdots and \lim_{k\to\infty} \lambda^*(A_k) < \lambda^*\Big(\bigcup_{k=1}^{\infty}...

Homework Statement Define the Fibonacci Sequence as follows: f1 = f2 = 1, and for n≥3, $$f_n = f_{n-1} + f_{n-2}, $$ Prove that $$\sum_{i=1}^n f^{2}{}_{i} = f_{n+1} * f_{n} $$ Homework Equations See above. The Attempt at a Solution Due to two variables being present in both the Sequence...

Hello all, To my knowledge, there are still a lot of questions regarding the specifics of black hole formation. My question is in regards to formation time. I've read that the actual formation takes "less than a second" according to the equations. Does anyone know where time shows up in the...

In the eponymous short story Lunarians have a large dome they can use to fly under their own power, using wings they rent. BUT, let's posit a person who regularly runs 26.2 marathons on Earth. Is this person fit/strong enough to actually fly in that arena? SWAGs accepted.

Why nature has a lot of application of fibonacci sequence I mean why the number of spirals in the head of sunflower always has to be a memeber of fibonacci sequence, why pinecones displays similar patterns and many more examples. Do we really know the answer? I mean is this question is...

Homework Statement Find the limit of the following sequence: ##L_2 = \lim_{n \rightarrow +\infty} \frac {\sum_{k=0}^n (2k - 1)^p}{n^{p+1}}## Homework Equations 3. The Attempt at a Solution [/B] Seeing that ##\lim_{n \rightarrow +\infty} n^{p+1} = + \infty ## i can apply the Stolz theorem. (Is...

First post and I'm wondering if I could get some help. I'm new to queueing theory so I'm not sure how to solve this problem. Population-------------> Queue--------------> Server I have a calling population that is infinite or a vast amount. The queue capacity is limited at 13. There is 1...

Question: Find the first 5 terms of this series and determine if it is an arithmetic sequence. An= 2 + 6n Help please!

I wanted to prove that the Fibonacci sequence is a divisibility sequence, but I don't even know how to prove it. all I know is that $$gcd\left({F}_{m},{F}_{n}\right)={F}_{gcd\left(m,n\right)}$$ and I should somehow use the Euclidean algorithm?

Need help with a homework question! The question gives: The first three terms of a geometric sequence are sin(x), sin(2x) and 4sin(x)cos^2(x) for -π/2 < x < π/2. First I had to find the common ratio which is 2cos(x) Then the question asks to find the values of x for which the geometric series...

{\displaystyle \sum_{n=1}^{\infty}a_{n}} is converage, For N\in \mathbb{N}\sum_{n=N+1}^{\infty}an is also converage proof that \lim_{N\rightarrow\infty}(\sum_{n=N+1}^{\infty}an)=0 {\displaystyle \sum_{n=1}^{\infty}a_{n}} is converage, For N\in \mathbb{N} \sum_{n=N+1}^{\infty}an is...

Hi, I am studying a positive sequence detector and have some trouble understanding it. I understand most of it, but the author in the book "Instantaneous Power Theory and Applications to Power Conditioning" writes: For extracting the fundamental positive-sequence voltage with the dual method...

I recently read about zero sequence braking of induction motor. In zero sequence braking of induction motor, the stator phases are connected in series and a single phase ac supply is connected across the series combination. Thus, the currents in all three phases are co-phasal, which are called...

I am stuck in the middle of building a circuit for an aircraft starting mockup that will train our students to correctly start the aircraft without accidentally re-engaging the starter and thus causing very costly maintenance actions. The system is 12v. We would like to have a 10 second delay...

If a protein contains a repetitive region, what might be assumed, and what should be done next to test the null hypothesis? Can anyone answer this with a reliable source they find?

I did 15+30+60+120+240+480+960 to get 1905, then continued the sequence to get 4. Is this right?

To measure negative sequence impedance of an alternator, we short the white and blue phases, measure the current and then divide it by the voltage of the red phase. Then to measure zero sequence impedance, we short the rotor and connect the stator in open delta and inject a current. My...

So the definition of a bounded sequence is this: A sequence ##(x_{n})## of real numbers is bounded if there exists a real number ##M>0## such that ##|x_{n}|\le M## for each ##n## My question is pretty simple. How does one choose the M, based on the sequence in order to arrive at the...

Hello Let ##(x_n)_{n=1}^\infty## be a real sequence and ##L \in \mathbb{R}##. Consider the following conditions on ##(x_n)_{n=1}^\infty## and ##L##. $$\forall \varepsilon > 0,~ \forall n_0 \in \mathbb{N},~\exists n \in \mathbb{N}\mbox{ so that } (n \ge n_0 \mbox{ and } |x_n - L| < \varepsilon)...

Homework Statement I am trying to wrap my head around what it means to find an explicit formula for the sequence of partial sums. Question: Find an explicit formula for the sequence of partial sums and determine if the series converges. a) sum from n=1 to n=infinity of 1/(n(n+1)) Homework...

Homework Statement Consider the space ##([0, 1], d_1)## where ##d_1(x, y) = |x-y|##. Show that there exists a sequence ##(x_n)## in ##X## such that for every ##x \epsilon [0, 1]## there exists a subsequence ##(x_{n_k})## such that ##\lim{k\to\infty}\space x_{n_k} = x##. Homework Equations N/A...

Homework Statement Give the example and show your understanding: [1][/B].Lets define some property of a point of the function: 1. Point is a stationary point 2. Point is a max/min of a function 3. Point is a turning point of a function If possible name a function whose point has properties of...

Homework Statement Either give an example or show that no example exists. An unbounded sequence for which 3 is an upper bound, and no β < 3 is an upper bound. Homework EquationsThe Attempt at a Solution example: {k}3k=-∞ and by this notation i mean that k starts at -∞ and ends at 3, and k∈ℤ...

Homework Statement [/B] Hi Theorem attached and proof. I am stuck on 1) Where we get ##|g(z)|\geq |a_m|/2 ## comes from so ##a_{m}## is the first non-zero Fourier coeffient. So I think this term is ##< |a_m|r^{m}##, from ##r## the radius of the open set, but I don't know how to take care...

Homework Statement Suppose a_{n}=\frac{n^2-2n+1}{2n^2+4n-1} For each positive number \epsilon , find a number N such that: \mid a_{n} - L\mid < \epsilon whenever n > N. Homework EquationsThe Attempt at a Solution \mid \frac{n^2 -2n + 1} {2n^2+4n-1} - \frac{1} {2} \mid < \epsilon...

I am looking at a statement that, for a short exact sequence of Abelian groups ##0 \to A\mathop \to \limits^f B\mathop \to \limits^g C \to 0## if ##C## is a free abelian group then this short exact sequence is split I cannot figured out why, can anybody help?

Homework Statement Clearly if a sequence of points ##\{x_n\}## in some space ##X## with some topology, then the sequence will also converge when ##X## is endowed with any coarser topology. I suspect this doesn't hold for endowment of ##X## with a finer topology, since a finer topology amounts...

Homework Statement Given the sequence ##\frac{1}{2}(x_n + \frac{2}{x_n})= x_{n+1}##, where ##x_1 =1##: Prove that ##x_n## is never less than ##\sqrt{2}##, then use this to prove that ##x_n - x_{n+1} \ge 0## and conclude ##\lim x_n = \sqrt{2}##. Homework EquationsThe Attempt at a Solution...

Homework Statement Let ##E \subseteq M##, where ##M## is a metric space. Show that ##p\in \overline E = E\cup E' \Longleftrightarrow## there exists a sequence ##(p_n)## in ##E## that converges to ##p##. ##E'## is the set of limit points to ##E## and hence ##\overline E## is the closure of...

I'm currently an Aerospace major/Physics minor who's interested in delving further into mathematics. What is the optimal sequence of topics that I should follow for self-studying. I know this forums has links to a lot of interesting books, but I don't really know which one to start with. Thanks...

Suppose I need to fill a row composed of 32 cells with a sequence of 4 cells with value 1 in a row followed by 4 cells with value 0 in a row and then again 4 cells of 1 and so on. How can I do this without copying those cells and repeating pasting them in their location?

Sorry for my ignorance... still trying to get to grips... If a lady in the middle of a moving train sends out beams to the front and back of the train. They reflect off mirrors back to her and arrive simultaneously because she can't do an experiment to give away that she is moving forward...

Problem: Let f ∶ V → V be a linear operator on a finite-dimensional vector space V . Prove that the sequence 0 → ker(f) → V → im(f) → 0 is exact at each term. Attempt: If I call: a: 0 → ker(f), b: ker(f) → V, c: V → im(f), d: im(f) → 0. Then the sequence is exact at: ker(f) if...

Hey! :o Let $(a_n)_{n=1}^{\infty}$ be a real sequence and let $(b_n)_{n=1}^{\infty}$ a sequence in the set of limit points of $(a_n)_{n=1}^{\infty}$, $L(a_n)$. There is also a $b_0\in \mathbb{R}$ with $b_n\rightarrow b_0$ for $n\rightarrow \infty$. I want to show that then $b_0\in L(a_n)$...

Is the series of numbers 2,3,5,8,13,21 ... a fibronacci sequence ? Because it doesn't start with 1 , but it fulfills the explicit formula .

I am currently picking out my courses for grade 11 next year. I have the opportunity to take both grade 11 AP Physics 1 and grade 12 AP Physics 2 in the same year (semestered). If I do I won't have any physics courses in grade 12 instead I'll be taking grade 11 Pre-AP Chemistry and grade 12 AP...

$\tiny{s4.11.1.26} \\$ $\text{ Determine whether the sequence converges or diverges. If it converges, find the limit.} \\$ $$\displaystyle a_n=\frac{(-1)^n n^3}{n^3+2n^2+1}$$ $\text{ divide every term by $n^3$}$ $$\displaystyle a_n=\frac{(-1)^n }{1+\frac{2}{n}+\frac{1}{n^3}}$$ $\text{ take the...

Homework Statement [/B]Homework Equations delta(x) = [b-a]/n xi=a+delta(x)i The Attempt at a Solution So, it is said that i have to use riemann sums to solve this one. what i did is i took the 1/12k out thus getting 1/12k[1/[1+1/12k] + 1/[1+2/12k] + 1/{1+19k/12k]] I found that xi =...

Homework Statement I am trying to design a 5-bit sequence recognizer. The circuit has to detect two sequences of bits. The first sequence is 11001. The second is 10010. The output when no sequence should be 00. When the 1st input sequence is detected, the output should be 01. When the second...