Sequences Definition and 576 Threads

Homework Statement Find the limit as n\rightarrow\infty of the sequence an=\frac{(2n)! 22n}{(n!)2 (2n+1) 52n} Homework Equations The Attempt at a Solution

Homework Statement Let f: N -> N be a bijective map. for n Є N a sub n = 1 / f(n) Show that the sequence (a sub n) converges to zero. Homework Equations The Attempt at a Solution Basically I have been stuck on this problem for hours now and have read through my notes and...

Homework Statement Prove that if two sequences an and bn converge to L and M respectively, then the sum of the sequences converge to L+M. Present a counter example to show the sum of two divergent sequences need not be divergent. The Attempt at a Solution We have an -> L and bn -> M we want an...

If a_n >= 0 for all n, and the series a_n converges, then n(a_n - a_n-1) --> 0 as n --> infinity. Prove or disprove the statement using a counterexample. I know that the statement is false...I am just having terrible difficultly finding a counterexample...

Struggling with this topic! :( got a couple of questions. Homework Statement 1) Determine the value of the improper integral when using the integral test to show that \sumk/(e^k/5) is convergant given answers are a)50/e b)-1/(5e^1/5) c)5 d)5e e)1/50e 2) determine whether \sum...

Homework Statement Determine whether the sequence is convergent or divergent. Find limits for convergent sequences. c_{1} = 4, c_{n+1} = -\frac{c_{n}}{n^{2}} for n \geq 1 Homework Equations lim_{n\rightarrow\infty} a_{n} = L Where L is a number. The Attempt at a Solution...

Hello! I have just another problem I can't figure out how to solve: Homework Statement Consider a homomorphism of short exact sequences (it's all vector spaces): [PLAIN]http://img814.imageshack.us/img814/9568/seq.png Prove that: (1) \sigma is surjective iff \rho is injective. (2) \sigma is...

Homework Statement If {cn} is a convergent sequence of real numbers, does there necessarily exist R> 0 such that |cn|≤ R for every n ∈ N? Equivalently, is {cn : n ∈ N} a bounded set of real numbers? Explain why or why not. Homework Equations n/a The Attempt at a Solution I would...

Homework Statement \lim_{n\to\infty}a_n=l \rightarrow \lim_{n\to\infty}\frac{a_1+\dots+a_n}{n}=l Homework Equations N/A The Attempt at a Solution Could someone verify that this proof works? I would really appreciate it. Proof: Since the sequence \{a_n\} converges to l, for any...

Homework Statement Give an example of a sequence {an} whose value is 7 for infinitely many values of n, but which does not converge to 7 Homework Equations The Attempt at a Solution I tried to think about such a sequence but cannot come up with any that satisfies that its value...

\mathbb R can be defined as "any (Dedekind-)complete ordered field". This type of abstract definition is a different kind than e.g. the "equivalence classes of Cauchy sequences" construction. I prefer abstract definitions over explicit constructions, so I would be interested in seeing similar...

CAN U ALL HELP ME TO SOLVE THIS QUESTION? I don't know how to start... The sum of the first 2n terms of a series P is 20n-4n2. Find in terms of n, the sum of the first n terms of this series. Show that the series is an arithmetic series.

Homework Statement Scanned and attached Homework Equations I am guessing a combination of induction and the telescoping property. The Attempt at a Solution I'm studying this extramurally, and I've just hit a wall with this last chunk of the sequences section, so if someone can...

I don't understand anything about this question: In a pest eradication program, N sterilized male flies are released into the general population each day, and 90% of these flies will survive a given day. A) Show that the number of sterilized flies in the population after n days is N +...

Homework Statement The sum of the first six terms in a geometric sequence of real numbers is 252. Find the sum of the first four terms when the sum of the first two terms is 12. Homework Equations Sn = A1 - A1Rn divided by 1 - R R \neq 1 (I can't figured out how to make the...

Homework Statement Evaluate lim sqrt(n)*[sqrt(n+1)-sqrt(n)] Homework Equations sqrt(n)/[sqrt(n+1)+sqrt(n)] = 1/sqrt[1+(1/n)]+1 The Attempt at a Solution I know that limit sqrt(n) = Infinity and that limit (sqrt(n+1)-sqrt(n)) = 0. And I know that sqrt(n)*(sqrt(n+1)-sqrt(n)) =...

Let f_n : [0,1] → [0,1] be a sequence of Riemann integrable functions, and f : [0, 1] → [0, 1] be a function so that for each k there is N_k so that supremum_(1/k<x≤1) of |f_n(x) − f(x)| < 1/k , for n ≥ N_k . Prove that f is Riemann integrable and ∫ f(x) dx = lim_n→∞ ∫ f_n(x) dx I am really...

Homework Statement Well, my problem is proving that sequences are in fact Cauchy sequences. I know all the conditions that need to be satisfied yet I cannot seem to apply it to questions. (Well, only the easy ones!) My question is, prove that X_{n} is a Cauchy sequence, given that...

Homework Statement Find the limit: an= 2n/(n2+1)1/2 Homework Equations n/aThe Attempt at a Solution Because n is approaching infinity, is it OK to disregard the +1 in the denominator and just consider the denominator to be n? This would then divide out the n in the numerator leaving 2...

Homework Statement Express 7.54545454545 . . . as a rational number, in the form p/q where p and q are positive integers with no common factors. p = ? and q = ? This problem is nothing like I've seen before, so I don't even have a clue on how to start it.

Suppose that ak is a decreasing sequence and (ak) approaches 0. Prove that for every k in the natural numbers, ak is greater than or equal to 0. I was thinking I should assume the sequence is bounded below by 0 and do a proof by contradiction. Any suggestions?

Homework Statement Let sum of a sub k be an absolutely convergent series. a. Let f be the function defined by f(x) = sum of (a sub k) * sin(kx). Prove that: the integral from 0 to pi/2 of f = sum of (a2k-1 + a4k-2)/(2k-1) Homework Equations I already showed that f(x) converges...

Homework Statement [PLAIN]http://img204.imageshack.us/img204/946/helph.jpg Homework Equations The Attempt at a Solution I have not idea what to write.. i read the whole chapter twice and don't know where to start :S

Homework Statement Let a_n be defined recursively by a_{1}=1, a_{n+1}=sqrt(6+a_{n}) (n=1,2,3,...). Show that lim n->infinity a_{n} exists and find its value The Attempt at a Solution Observe that a_{2}=\sqrt{6+1}=\sqrt{7} > a_{1}. If a_{k+1} > a_{k}, then a_{k+2} = \sqrt{6+a_{k+1}} >...

Homework Statement Suppose that a_n \to L and b_n \to L. Show that the sequence a_1, b_2, a_2, b_2, a_3, b_3, ... converges to L.Homework Equations The Attempt at a Solution I don't know.. how come b_2 is repeated? Do I need do use some kind of epsilon type proof?

Homework Statement I need to see if these sequences converge or diverge: 1) a_n = ncosn\pi 2) { 0,1,0,0,1,0,0,0,1,0,0,0,0,1,... } 3) a_n = \frac{1 . 3 . 5 . ... (2n - 1)}{n!} Homework Equations The Attempt at a Solution 1) ]cosn\pi = -1[/itex] so [itex]a_n \to...

I want to express ...-2\pi+\theta,\theta, 2\pi-\theta, 2\pi+\theta, 4\pi-\theta... in terms of a variable integer k. e.g. ...-x,0,x, 2x, 3x... = kx, k E Z So I was thinking expressing it as so: 2k\pi \pm \theta but I believe it can be expressed in another way to avoid the \pm, using (-1)k...

Hi all, I was just wondering whether one could define arithmetic sequences in R^2 in a simmilar manner as in R.? Here is what i see as a natural way of doing it, but neither have i read about it, nor heard. \mbox{ Let } x_n \in R^2 \mbox{ be a sequence given as follows : } x_n=a+mb\\...

[SIZE="3"](a) lim_{n\rightarrow\infty} (\sqrt{(n + a)(n + b)} - n) where a, b > 0 [SIZE="3"](b)lim _{n\rightarrow\infty} ([SIZE="3"]n!)1/n2

Homework Statement Find the expression for the nth term for the sequence {1,1,-1,-1,1,1,-1,-1,...} The Attempt at a Solution No idea.

Is the following statement true or false, and why: In an infinite non repeating number sequence (like the digits of pi), any given finite number sequence will appear in it.

How many heads in a row would you expect to find if you toss a fair coin 8 times? I am thinking that the probability of 3 heads is 1/8 and since you have 8 tosses, that would give an E(x) of 1. So I am guessing that 3 is the number of heads one would expect to see in a fair coin...

Homework Statement Given R is complete, prove that R2 is complete with the taxicab norm The Attempt at a Solution you know that ,xk \rightarrow x , yk \rightarrow y Then, given \epsilon, choose Nx and Ny so that \left|x_n - x_m\left| and \left|y_n - y_m\left| are less than...

Homework Statement x_{n}(t) \left\{\begin{array}{cc}nt,&\mbox{ if } 0\leq t \leq \frac{1}{n}\\ \frac{1}{nt} & \mbox{ if } \frac{1}{n}\leq t \leq 1 \end{array}\right. Homework Equations The Attempt at a Solution Can someone help me get started finding the limit as n -> inf...

Homework Statement This is the Theorem as stated in the book: Let S be a subset of a metric space E. Then S is closed if and only if, whenever p1, p2, p3,... is a sequence of points of S that is convergent in E, we have: lim(n->inf)pn is in S. Homework Equations From "introduction to...

Homework Statement Let {an} be a sequence of real numbers. Suppose an->L as n->∞. Prove that [(a1+a2+...+an)/n] ->L as n->∞. Homework Equations N/A The Attempt at a Solution By definition: an->L iff for all ε>0, there exists an integer N such that n≥N => |an - L|< ε. Given ε>0...

Homework Statement But I think the definition is as follows: Let an be a sequence of real numbers. Then an->a iff for ALL ε>0, there exists an integer N such that n≥N => |an - a|< ε. The definition says that it has to be true for ALL ε>0, but in the example above, they just let ε to...

Homework Statement Definition: Let an be a sequence of real numbers. Then an->a iff for all ε>0, there exists an integer N such that n≥N => |an - a|<ε. [for all of the following, "lim" means the limit as n->∞] Theorem: Suppose lim an =a and lim bn =b. Then lim (an + bn) = a + b. Proof...

Homework Statement Definition: Let an be a sequence of real numbers. Then an->a iff for all ε>0, there exists N such that n≥N => |an - a|<ε. Let an=(n2+1)/(n2-9). PROVE that an->1 as n->∞. Proof: Assume n≥4. Then | 1-an | = 10/(n2-9). 10/(n2-9) < 10/n provided n2 - 9 > n, i.e. n2 - n...

Given positive interger N, now many non-decreasing sequences of length N are there whose entries are less than N?

Hi, I'm looking to show that the metric space of convergent complex sequences under the sup norm is not separable; that is what I assume it is since I cannot find a way to prove that it is separable (I am unable to find any dense subsets). A set of complex sequences convergent to a certain...

Homework Statement Prove that every non-decreasing, bounded sequence of rational numbers converges to some real number using Dedekind cuts. Homework Equations A real number is a set \alpha, of rational numbers, with the following four properties: If x \in \alpha and y is a rational...

Homework Statement I have a function a_{n}=\frac{2+3n}{2n+1}, and I have to find out whether it converges or diverges. I did the ratio test lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|. And according to the divergence test, it should diverge. Then it asks if the series...

Homework Statement Let Sn be a sequence in R Prove lim Sn= = 0 if and only if lim abs(Sn) = 0 Homework Equations none The Attempt at a Solution I think this is someone ciruclar logic and that is why I am stuck Assume lim Sn = 0, thus for n > N implies |Sn| < epsilon or...

Homework Statement Find x so that x+5, 3x+1, and 4x+1 are consecutive terms of an arithmetic sequence. Not really sure how to do the problem at all. Some assistance would be much appreciated.

Homework Statement An = (n+3)^1/(n+3) Converge or diverges? Find the limit of the convergent sequences. Homework Equations The Attempt at a Solution I have the solution but I don't understand it. I'm looking to get it. It shows me taking the limit as x->infinity of x^1/x and...

Homework Statement Let (rn) be an enumeration of the set Q of all rational numbers. Show that there exists a subsequence (rnk) such that limk\rightarrow\infty rnk = +\infty Homework Equations The Attempt at a Solution Im not sure how to even attack this

I have a test on this stuff and I'm confused about some things. First, how do I show work to this problem? I know that its absolute value diverge but I don't know how to show the work. Also not sure how to show the work for alternating series test. Not sure how to show that the it's...

I know that for any two real sequences x_n and y_n, we have \liminf_{n\to \infty} x_n + \liminf_{n\to \infty} y_n \leq \liminf_{n\to \infty} (x_n + y_n). I also know that, if one of the sequences converges, the inequality becomes equality. My question is this: If I've managed to show that...

Homework Statement For the sequence (see picture), explore its monotones, define supermum and infimum, minimum and maximum, and find if the sequence is convergent. If the sequence is convergent, find form where on the terms of this sequence differ from the limit for less than ε = 0.01. The...