Sequence Definition and 1000 Threads

Let $a_1 = 1$, $a_2 = a_3 = 2$, $a_4 = a_5 = a_6 = 3$, $a_7 = a_8 = a_9 = a_{10} = 4$, and so on. That is, $a_n ∶ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, . . . . $ What is $a_{2013}$?

Let the sequence {$a_n$} be defined by $a_1 = 1$ and $a_n = \frac{1}{1+a_{n-1}}$ for $n \ge 2$ . Find the limit.

Assume the sequence of positive numbers ${a_n}$ converges to L. Prove that $\lim_{n \to \infty} \sqrt[n]{a_1a_2...a_n} = L$ (The nth root of the product of the first n terms) Since ${a_n}$ converges we know that for every $\epsilon> 0$ there is an $N$ such that for all $n > N$ $ |a_n -...

determine whetehr the sequence diverges or converges, if converges find its limit \displaystyle a_n = (1 + \frac{2}{n})^n my teacher is trying to tell me this converges to e^2 ... HOW?

If ##\left\{ a_{n} \right\}## is monotone increasing and there exists ##M \in \Re## such that for every ##n \in N## ##a_{n} ≤ M## prove that ##\left\{ a_{n} \right\}## converges. (Hint: Use the Cauchy sequence property. Recall: 1) ##\left\{ a_{n} \right\}## is Cauchy if and only if...

a sequence ${a_n>0}$ for all n given :$S_n - \dfrac{1}{a_n}=a_n-S_n$ find :$S_{2013}^2$ where :$S_n=a_1+a_2+-------+a_n$

Determining if the sequence converges or diverges, if it converges find the limit \sqrt{n^2 + n} - n Wouldn't this just diverge if n--> infinity ? I'm not sure what to do here? I can;t use lopitals...Also how would this converge to 1/2 is this a telescoping series?

Determine if the sequence converges or diverges, if it converges find the limit n sin\frac{1}{n} so what I did was \frac{sin(1/n)}{1/n} and then then took the limit as n --> infinity and got 1...Which I guess i really didn/t need to divide by 1/n but oh well.. Would it then be correct to say...

I'm confused on how they are getting their result... Determine if the sequence converges or diverges, if it converges, find the limit... \frac{n^2}{2n - 1} - \frac{n^2}{2n + 1} So I started plugging in from 1 because it looks like they want me to do something with a telescoping series and I...

Determine if the sequence converges ot diverges. If it converges, find the limit \frac{tan^{-1} n}{n} So I'm thinking that I can say tan inverse is \frac{\frac{\pi}{2}}{n} as n--> infinity is going to be some number over infinity = 0? so therefore it converges to 0?

I am trying to convert a FORTRAN version of a Canonical Correlates program listed in Multivariate Morphology - Blackith and Reyment. I've programed in FORTRAN decades ago and now I have to understand the language to rewrite the program into PERL. a) DO 260 I = 1 , M b) SX( I ) =...

Can someone check my solutions? Do the following sequences {a_n} converges or diverge as \n\to\infty? If a sequence converges find its limit. Justify your answers. 1. a_n = 2 +(-1)^n Answer: so can I say that as lim n --> infinity the sequence diverges by oscillation? 2. a_n = \frac{n}{e^n}...

Let $R$ be a commutative ring and $0\to L\to M\to N\to 0$ be a sequence of $R$ modules. Let $A$ be a multiplicativity closed subset of $R$ so that we can consider the corresponding localisation sequence: $0\to A^{-1}L\to A^{-1}M\to A^{-1}N\to 0$. Suppose that the localisation sequence is exact...

the sequence: 1,1,2,1,1,2,1,1,3,1,1,2,1,1,2,1,1,3,1,1,2,1,1,2,1,1,4, "............" 4, "............" 5, "............" 4, "............" 4, "............" 5, "............" 4, "............" 4, "............" 6, and so on, is the sequence of exponents of 3 in the prime factorization of...

Homework Statement Let \{P_i\}_{i=0}^\infty be a sequence of points on a plane. Suppose P_is are placed as on the picture below, so that |P_0 P_1|=2, |P_1 P_2|=1, |P_2 P_3|=.5, |P_3P_4|=.25, ... Find the coordinate of the point P = \lim_{i→\infty} P_i Homework Equations The Attempt...

Homework Statement This is a 3 state machine with one input variable. The input given for x produces the output sequence for z. The machine starts in state A. I am asked to derive the state table. x=010001010010010011010 A z=001000000001001000001 Homework Equations The Attempt...

Hey! ;) I am looking at the following exercise: Find a sequence of differentiable functions $f_n$,such that $f_n \to f$ uniformly,where $f$ is differentiable, $f_n' \to g$ pointwise,but $f'\neq g$. How can I find such a sequence of functions? Is there a methodology to do it?? :confused:

Hi everyone, After completing an undergraduate engineering degree, I walked away with a feeling that all I was taught was to crunch numbers, lacking an intuitive understanding of solution mechanisms. Now, with spare time, I got the desire to re-learn my upper mathematics curriculum. One of...

Homework Statement Problem is attached in this post Homework Equations Problem is attached in this post The Attempt at a Solution Lim n(2^(1/n)-1) as n -> ∞ Lim (2^(1/n)-1)/(1/n) as n -> ∞ -> 0/0 -> Indeterminate Form -> L'Hopital's Rule However, I can't seem to figure...

Hey! :o An infinite orthonormal system $\{e_1, e_2, ... \} \subset H$ is closed in $H$ iff $\forall x \in H$ $$||x||^2=\sum_{i=1}^{n}{|(x,e_i)|^2}$$ From the summability of the right part of the relation above, we conclude to that the sequence $(x,e_i)$ is a zero sequence. Could you explain...

Find the number of different terms of the finite sequence $\left\lfloor \dfrac{x^2}{1998} \right\rfloor$ where $x=1,\,2,\,3,\,\cdots,\,1997$.

Homework Statement Consider the sequence {an}\subsetR which is recursively defined by an+1=f(an). Prove that if there is some L\inR and a 0≤c<1 such that |\frac{a_{n+1}-L}{a_{n}-L}|<c for all n\inN then limn\rightarrow∞an=L. Homework Equations Definition of convergence: Suppose (X,d) is...

Homework Statement Define f_n : \mathbb{R} \rightarrow \mathbb{R} by f_n(x) = \left( x^2 + \dfrac{1}{n} \right)^{\frac{1}{2}} Show that f_n(x) \rightarrow |x| converges uniformly on compact subsets of \mathbb{R} Show that the convergence is uniform in all of \mathbb{R}...

Homework Statement Prove for c>0 the sequence {x_n} = \frac{1}{2}(x_{n-1} + \frac{c}{x_{n-1}}) converges. The Attempt at a Solution This is proving difficult, I have never dealt with recursive sequences before. Any help would be appreciated. Thanks.

Consider the sequences $(c_n)_n,\,(d_n)_n$ defined by $c_0=0$, $c_1=2$, $c_{n+1}=4c_n+c_{n-1}$, $n \ge 0$, $d_0=0$, $d_1=1$, $d_{n+1}=c_n-d_n+d_{n-1}$, $n \ge 0$. Prove that $(c_n)^3=d_{3n}$ for all $n$.

Homework Statement create an algorithm for a Fibonacci sequence that will return the f value in f(n) = f(n-1)+f(n-2) Homework Equations f(n) = f(n-1)+f(n-2) The Attempt at a Solution I have tried several ways to create an algorithm that will sum the two previous numbers but always end up...

Hi MHB, My nephew, age 9, was asked the following question and he hoped I could solve the problem and then explain the solution to him using only elementary math concepts. My boyfriend has solved it, but he used a formula that he recalled seeing in a textbook by G.H. Hardy, and that...

Homework Statement consider the information given below about three main sequence stars. Star 1 will be a main sequence star for 4.5 billion years. Star 2 has a spectral type of M5. Star 3 has the same luminosity as the Sun. Which has the greatest mass or are they approx. the same?Homework...

Hi y'alllll! Need some guidance on a little issue, As you very well know, european standards (ecss) exist for space systems but they are generally outlined for launch vehicles. Do specific standards exist for payload fairings? For example a test sequence is outlined for general space...

For a system I am studying the following sequence (which I would assume is quite common) came up: n1=1, n2=2, n3=4, n4=7, n5=11, n6=16, n7=22 ... i.e. the difference betweens two successive numbers grows with 1 as we move from (n_N-1, n_N) to (n_N,n_N+1). Is there a closed form expression f(k)...

Hello, I am looking for an help about this, I have very short time to do many of them and those are an example, could someone show me one solution or explain me how to do it? Thank you if you can help me, I really appreciate. Francesco.

Hey again! (Blush) I have to check if the sequence $a_{n}=\frac{1}{\sqrt{n^2+1}}+\frac{2}{\sqrt{n^2+2}}+...+\frac{n}{\sqrt{n^2+n}}$ converges.I thought that:$$\frac{n^{2}(n+1)}{2\sqrt{n^2+n}} \leq a_{n} \leq \frac{n^{2}(n+1)}{2\sqrt{n^2+1}}$$ Because of the fact that: $$\lim_{n \to...

Define the sequence of integers a1, a2, a3,... as follows: a1 = 3 a2 = 6 an = 5an-1 - 6an-2 + 2 for all n ≥ 3 Prove that an = 1 + 2n-1 + 3n-1 ------------------------------------------------------------------------------------------------ my attempt: base case: n=1 1+ 20 +30 = 1...

Hey! I want to check if the sequence $a_{n}=\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+...+\frac{1}{\sqrt{n^2+n}}$ converges. I thought that I could find the difference $a_{n+1}-a_{n}$ to check if $a_{n}$ is increasing or decreasing.I found...

Homework Statement In a question I was asked, assuming a spectrometer reading of Hydrogen produced two strong spectral lines at 656.3nm and 410.1nm. And also assuming the diffraction grating had 500 lines/mm What is the highest order of spectrum which can be fully observed , i.e value of m...

Problem is: "By experimenting with numerous examples in search of a pattern, determine a simple formula for (F n+1)^2-(F n-1)^2; That is, a formula for the difference of the squares of two Fibonacci numbers." The n+1 and n-1 should be smaller by the F but I don't know how to do that on a...

Hey! :) I have a question. It is given that $a>0 , x_{1}=x>0 \text{ and } x_{n+1}=\frac{1}{2}(x_{n}+\frac{a}{x_{n}})$ and I have to show that the sequence $(x_{n})$,at least from its second term,is decreasing and bounded from below from $\sqrt{a}$.Also,I have to find the limit $\lim_{n \to...

Hi, Suppose we're looking at a random sequence of digits from 0 to 9. We start off reading the digits until every digit from 0 to 9 has been seen at least once and we mark the count of digits read up to that point (run length). We then reset the run length and continue until the whole random...

Homework Statement Check whether the sequence a_{1}=\alpha ,\alpha > 0, a_{n+1}=6*\frac{a_{n}+1}{a_{n}+7} converges and find its limit if it does, depending on α. Homework Equations The Attempt at a Solution I showed boundedness([0,6]) and found that in the case of convergence...

Homework Statement Given: x_{n+1}=\frac{1}{3+x_n} with x_1=1 Show that: (1) |x_{n+1}-x_n| \leq \frac{1}{9}|x_{n}-x_{n-1}| and (2) x_n is Cauchy. Homework Equations The Attempt at a Solution I've tried different approaches (including induction) but the...

Homework Statement Given the sequence: a(1) = 1, a(n+1)=0,5(a(n)+2/a(n)) n>=1 Homework Equations I have found through speculations that the limit value is SQRT(2). The Attempt at a Solution I started by proving that for n>1; a(n+1) < a(n) and also proved that for n>1 the...

Homework Statement Consider the sequence ak = (3)/(4^(2k)). Show is convergent, find sum. Please check work. Homework Equations ak = 3/(4^2k) let s {n} be the series associated with the sequence. Cannot write summation notation here, but k starts at 1 (k = 1 on bottom) and infinity...

Hello again, I am having trouble with a particular limit problem and would appreciate any help/pointers you can offer. The question is asking for the nth term of the sequence 2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4} .. and also asks for a limit of the sequence. My immediate guess was to apply...

Homework Statement "In a geometric sequence, the sum of t7 and t8 is 5832, the sum of t2 and t3 is 24. Find the common ratio and first term." Homework Equations d = t8/t7 or t3/t2 tn = a * rn-1 The Attempt at a Solution So I thought of developing a system of equations then solving...

I'm looking for 1-2 rigorous books on convex analysis for someone who already has some exposure to convexity, linear and nonlinear programming in an applied course. It seems that Rockafellar (Convex Analysis) and Fenchel (Convex Cones, Sets and Functions) is the classic treatment. Is there a...

$x_0,x_1,-----,x_{2004} \in Z , \, x_0=0, \mid x_n \mid =\mid x_{n-1}+1\mid $ $for, \,\, 1 \leq n \leq 2004$ (1) $find :\,\, min\mid x_1+x_2+x_3+ ------+x_{2004}\mid $ (2) get a set of numbers $ x_1,x_2,-----x_{2004} $ satisfying your answer

Problem: The first six values of the 10-point DFT of a real-valued sequence x(n) are given by {10, −2 + j3, 3 + j4, 2 − j3, 4 + j5, 12} Determine the DFT of x[n] = x[n+5] (10 point sequence) Relevant Equations: DFT(x[n-m]) = exp(-j*(2pi/N)*k*m) * X(k) where N = 10 ; m = -5...

Dear Forum, Does every sequence have a general term? What is the definition of the general term of a sequence? Best wishes

Using the steps below, show that the following sequence converges: 1+\frac{1}{2}-\frac{2}{3}+\frac{1}{4}+\frac{1}{5}-\frac{2}{6}+\frac{1}{7}+\frac{1}{8}-\frac{2}{9}+\frac{1}{10}+\frac{1}{11}-\frac{2}{12}++-++-... i. Consider the subsequence (s2,s3,s5,s6,s8,s9,...) of the sequence of partial...

Consider the following sequence: a1 = p, where p is a prime number. an+1 = 2an+1 Prove there is no value of p for which every an is a prime number, or make me look dumb and construct a counterexample.