Sequence Definition and 1000 Threads

Homework Statement Find sequence (a_n) s.t. \lim_{N\rightarrow \infty} \sum_{n=1}^{2N} a_n and \lim_{N\rightarrow \infty} \sum_{n=1}^{2N+1} a_n both converge but \sum_{n=1}^{\infty} a_n diverges. I have no idea where to start to be honest. I'm confused at how this is possible. Isn't it always...

hi all, i asked this question on y! answers and one guy gave me an advice to come here, sign up, and ask. i hope I'm in the right place! i'm a freshman majoring in mathematical econ and minor in statistics to prepare myself for the actuarial exams (and open up other career options). for this...

A finite geometric sequence has t1 = 0.1024 and t2 = 0.256. How many terms does this sequence have if its middle term has a value of 156.25? My Solution Common Ratio: T2/T1=(.256)/(.1024)=2.5 What term # is the middle term? tn=ar^n-1 a=0.1024 r=2.5 tn=156.25...

I know that the sequences meets the following: (n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n) I've got the feeling that this sequence is alternating or decreasing, but I was unable to prove it. Usually I use induction to prove things about such inductive sequence but in this case I don't have real values t o...

I found this interesting exercise on a topology book I'm reading, but I don't have a clue what to do. Show that there is no sequence {g_n} of continuous functions from R to R such that the sequence {(g_n)(x)} is bounded iff x is rational (where R = set of real numbers).

Instead of using a sieve to remove non-primes from the sequence. 6x-1 x =0 to x=n 6x+1 x=0 to x= n What if you calculate and remove the non-primes. I have determined how to calculate the non-primes in this set. By subtracting them from the entire set you are left with all primes. I find this...

Homework Statement let (X,d) be a metric space and let A be a dense subset of X such that every Cauchy sequence in A converges in X. Prove that (X,d) is complete. Homework Equations (X,d) is complete if all Cauchy sequences in X converge. A is a dense subset of X => closure(A) = X...

It's been a while since I've done rate of convergence problems, how would i find the rate of convergence for either of these sequences? 1) limn->infsin(1/n)=0 2)limn->infsin(1/n^2)=0

Homework Statement I have to find all the partial limits {I hope this is how this term named in English} of a sequencesHomework Equations a_1=0 a_{2n}=\frac {a_{2n-1}} {3} a_{2n+1} = 1/3 + a_{2n} The Attempt at a Solution I have tried to prove first that sequences of all the even terms...

help me i don't know where i must to posting this task if 8+4=12 7+2=9 2+3=6 6+5=10 9+5 = ?

Homework Statement I need a example of a continuous function f:(X, d) -> Y(Y, p) does NOT map a Cauchy sequence [xn in X] to a Cauchy sequence of its images [f(xn) in Y] in the complex plane between metric spaces. Homework Equations If a function f is continuous in metric space (X, d), then...

if you have a sequence of events {A_1, A_2, ...} then an expression for the event that "infinitely many A_i's occur" is: U(n = 1 to inf, U(m = n to inf, A_m) ) but wouldn't U(n = 1 to inf, A_n) also satisfy this?

x_n = (n^2 / (n^2+1) , 1/sqrt(n)). prove that this sequence converges and find the limit. so as n approaches infinite it is clear that x_n approaches (1,0). so using the definition of a convergent sequence, i pick an epsilon > 0 and i have to find some N such that when n>N, sqrt( 1/(n^2 +...

Homework Statement Find \displaystyle\lim_{x\to\infty}x\sin(\frac{1}{x}) where x is a natural number, and this is a sequence, not a real function.Homework Equations The Attempt at a Solution I know the answer is 1, and that supposedly one solution is to introduce a dummy variable y = 1/x...

Homework Statement So the Fibonacci Sequence is defined by an = an-1 + an-2 a1=1, a2=1 We are more interested in the sequence of ratios of subsequent terms of the Fibonacci sequence define rn = an+1 / an How do we prove that.. rn = 2 - 1/(rn-2 + 1) for all n>2...

Homework Statement What is the limit of { \frac{n+1}{2n} } as n --> oo. Prove your answer. The Attempt at a Solution This is example from my book. Here is the problem: *I am using the capital letter "E" instead of "ε" in latex-code. Intuitively, 1 is small relative to n as n gets large, so...

erm just a quick question, as i read something that didnt quite make sense to me anyway, our sun will stay in its current form for about another 5 billion years before inflating into a red giant, now i was just browsing wikipedia reading up about various things, and was reading the article about...

Just a check of my work please, The topic is series and sequence, Question: The sequence u1, u2, u3,... where u1 is defined by u1=2 and Un+1 =un+4 Find the nth term, un, of the sequence. I got the answer, u1=-1 u2=-1+4= 3 u3= 3+4 = 7 So: un = 4(n-1)-1 This seems to work but not sure if it...

Homework Statement Corey has take a job with an initial salary of $26,000 an annual raises of $1,250/ (a) What will his salary be in the 6th year? (b) How much money in total will Corey have earned after six years? Homework Equations an = a1 + (n - 1) d Sn = n / 2 (a1 + an)...

Homework Statement For the arithmetic sequence (2 - x), (-6 + 2x), (x + 2), solve for x and find t10. Homework Equations an = a1 + (n - 1) d The Attempt at a Solution Would I have to start off like this below::: an = a1 + (n - 1) d d = (-6 + 2x)-(2 - x) = (x + 2)-(-6 + 2x)

Find \sqrt{1+\sqrt{1+2\sqrt{1+3\sqrt{1+...}}}}

Bench Top's thread revived a number string curiosity that once stumped me. I wonder if anyone else saw anything like this before and can give the sequence a name. Description 1. The sequence comprises only numbers from 1 to 2n. 2. Each number from 1 to 2n appears once and only once. 3...

Question : An infinite sequence of independent trails is to be performed . Each trails resulting in a success with probability p and failure with probability 1-p . What is the probability that i) atleast 1 success occurs in the first n trails ; ii) exactly k success occur in the first n...

Let (xn) be a seq of real nos and let sn = x1+x2+x3+...+xn / n. prove that if if xn is bounded and monotone, then sn is also bdd and monotone. How can i got about this one.. ? I got it in the test today and i couldn't figure it out. only hint i could think of is how do i prove if xn...

Homework Statement Let x1 > 9000, and xn+1 = )2009xn + 2010)/2011 for n >1 show that (xn) converges and find its limit Homework Equations Definition of a limit, Monotone Convergence Theorem. The Attempt at a Solution Since xn+1 is monotone for n>1 and bounded, then it...

Homework Statement Given a sequence of rational numbers, defined inductively as follows: s1 = 1, sn+1 = sn/2 + 1/sn, n>=1 prove that 1<=sn<=2 forall n>=1 Homework Equations The Attempt at a Solution I've got the solution to this but I don't understand a certain part, I was...

I'm reviewing for my analysis exam, and am having trouble with this question: Let (fn) be the Fibonacci sequence and let xn=fn+1/fn. Given that lim (xn) = L exists, find L. I believe I know what technique I have to use. I think I have to find two subsequences of (xn), write one in terms of the...

Homework Statement Prove that {(-1)n} n>=1 does not converge Homework Equations The Attempt at a Solution If i define two subsequences, say {(-1)2n} = A and {(-1)2n+1} =B of that original sequence, then A converges to 1 and B converges to -1 ? Is this correct at all?

Homework Statement Suppose [a,b] is a closed interval (on R), and {xn}n>=1 is a sequence such that a) xn belongs to [a,b] b) lim as n--> infinity xn = x exists prove x belongs to [a,b] Homework Equations The Attempt at a Solution Well since any sequence is bounded, then...

Homework Statement suppose that {an} is a convergent sequence of points all in [0,1]. prove that lim an as n-->\infty is also in [0,1] Homework Equations for all\epsilon>0, \exists a natural number N such that for all natural numbers n, if n>N, then absolute value(an-L)<\epsilon The...

Homework Statement consider the sequence 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, ... for which numbers \alpha is there a subsequence converging to \alpha? Homework Equations none that i can think of... The Attempt at a Solution i think the \alpha values are 1 and 0...

Homework Statement We are given the real sequence x_n+1 = (x_n)^2 - 100 + sin(n), some x_0 Prove that if the sequence is bounded with positive numbers, then necessarily 10 <= x_n <= 11 for all n>=0. Homework Equations The Attempt at a Solution I tried induction and...

Homework Statement Im trying to prove that a sequence is cauchy using its definition. the sequence is 81+14n/50+31n. i know it converges to 14/31 using L'hopitals rule but the assignment is to use the definition of cauchy i have tried some things but I am not 100% sure how to start all...

Homework Statement Define a sequence (fn) from n=1 to infinity of functions on [0,1] by fn(t)=t^n does the sequence converge in (CL^2[0,1],||.||2) Homework Equations The Attempt at a Solution I am struggling on where to start. I am fairly new to the L2 space and so would just...

Homework Statement Find all positive integers n such that both n + 2008 divides n^2 + 2008 and n + 2009 divides n^2 + 2009 Homework Equations - The Attempt at a Solution I have no idea where to start... I'm not even sure it's a sequence and series question. If it is then I have...

Homework Statement Forgive my lack of LaTeX, not learned how to use it yet. Anyway, the problem is: Use the inequalities 1/(n+1) < ln(n+1) - ln(n) < 1/n to show that the sequence {xn} from n=1 to infinity defined by xn = 1 + 1/2 + ... + 1/n - ln(n) is strictly decreasing and bounded below...

Homework Statement Let Sn be a sequence such that |Sn+1-Sn|< 2-n for all n in the natural numbers Homework Equations The Attempt at a Solution I understand what it means to be cauchy but I'm not sure how to prove this particular sequence is cauchy. Please help!

Homework Statement For n\geq1 let 2an \leq an-1 + an+1 Prove that an convergesHomework Equations n/aThe Attempt at a Solution 2an+1 \leq an + an+2 an+2 \geq 2an+1 - an How do I proceed? Ratio test?

Hey. I have | tn - t | < M How can i show that t - M < tn thanks

Homework Statement Let g_n : R -> R be given by gn (x) := cos2n (x), does gn' converge uniformly? Homework Equations The derivative is as follows; -2nsin(x)cos2n-1, which I have found converges pointwise to the 0 function. Formal definition of Uniform Convergence; For all e>0...

If you are given a sequence of integers such as: An=xn+yn where x and y are integers. and n=0,1,2,3... how would one find the recurrence relation? I tried writing An+1 in terms of An but it doesn't come out neatly because it doesn't translate so well. And there are terms raised to the n+1...

Homework Statement Show that x_n = \frac{x_{n-1}}{2} + \frac{1}{x_{n-1}} converges. x_0 > \sqrt2 Homework Equations The Attempt at a Solution I am semi-stumped. I guess I can use the AM-GM on x_n to show that it is greater than root 2. Then show that x_n -...

Homework Statement I have to study the limit of a sequence which is defined as follows. I'm not looking for an answer, just a method of how to do it, or even what this notation means. An = { (\frac{n^{2}}{3n^{3}+1}, \frac{4n^{2}}{n^{2}+1} ] n even An = { (\frac{n^{2}}{6n^{2}-4}...

Homework Statement Hello, I don't entirely understand the question here. It is the limit of a sequence, however it is defined on what appears to be an interval, and is different if n is odd or even. Excuse the messy notation here, I don't know how to use LaTeX. Let the sequence {A_n : n>=1}...

Homework Statement Prove that the limit of 2n/n! = 0 The Attempt at a Solution I've already successfully proven that 2n < n! for every n>4, so for every value of xn greater than 4, we know that 0<xn<1. I understand how the sequence is behaving, but I'm having trouble putting it into...

Homework Statement So, I am to calculate limit of a sequence given by a formula: \sum^{n}_{i = 1} \sum^{i}_{j = 1} \frac{j}{n^3} The Attempt at a Solution I've tried to write down the sequence explicite and this is what I get: \frac{1}{n^3} + (\frac{1}{n^3} + \frac{2}{n^3}) + ... +...

Homework Statement Calculate the limit of a given sequence for n \rightarrow \infty: \frac{1 - 2 + 3 - ... + (2n - 1) - 2n}{\sqrt{n^2 + 1}}The Attempt at a Solution The correct answer seems to be -1. I've tried to apply the Stolz theorem but failed to compute \sqrt{(n + 1)^2 + 1} - \sqrt{n^2...

Homework Statement Prove that if x>0 and <n> is a sequence with lim\rightarrow \infty = x, then there is a real number N such that whenever n>N we have xn>0. The Attempt at a Solution I was trying to prove this by contradiction, but got stuck. Any hints or tips would be greatly...

Homework Statement I have been asked to determine convergence or divergence of a sequence given the nth term. If the sequence converges, find its limit. an = ln\sqrt{n} / n Homework Equations The Attempt at a Solution I had thought that the sequence would follow L'Hopital's...

Can someone post a link to that website where you type in a sequence of numbers and it tells you the name of and descries the sequence and shows a formula if one exists.