Sequences Definition and 590 Threads

Homework Statement The sum of the first 9 terms of an arithmetic series is 216. The first, third and seventh terms of the series form the first three terms of a Geometric pattern. Find the first term and the common difference of the arithmetic pattern. Homework Equations The Attempt at a...

Hi, I've been reading a textbook on set theory and came across Cantor's proof of the statement that the set of the infinite binary sequences is uncountable. However there is one thing that is not clear to me: The nth such sequence would be: An = (an,0,an,1,...), n = 0, 1, 2,... where...

Homework Statement Let ##x_k=k## for ##k \leq 31## and ##\displaystyle x_{k+1}=\frac{x_1+x_2+...x_k}{k}## for ##k \geq 31##. Also let ##y_k=x_k## for ##k \leq 31## and ##\displaystyle y_{k+1}=\frac{y_k+y_{k-1}+...y_{k-30}}{31}## for ##k \geq 31##. Now if ##z_k=y_k-x_k## for all ##k ε N##. Find...

I have had some introduction to set theory and have gone through calculus in a theoretical manner up through first and second order differential equations. However, we are now working on sequences (and series, but I find series to be less of a problem). There doesn't seem to be an easy way to go...

1. Finding the limit of the sequence: { an } = 5n^(2) / (n^(2) + 2) Homework Equations 3. what i did was : lim as (n -> Infinity) of function [5n^(2) / (n^(2) + 2)] Then factored out the constant: 5{lim as (n -> Infinity) of function [n^(2) / (n^(2) + 2)]}...

Hi I am playing around with recursive definitions of Lucas and Fibonacci sequences: I came across a relationship L0 + L1 + L2 + L3 ... Ln = sum(i = 0, n) Li = Ln+2 -1; Sorry for the horrible notation, but could anyone provide a counter example using an inductive approach? I get the...

Homework Statement Determine the convergence, both pointwise and uniform on [0,1] for the following sequences : (i) ##s_n(x) = n^2x^2(1 - cos(\frac{1}{nx})), x≠0; s_n(0) = 0## (ii) ##s_n(x) = \frac{nx}{x+n}## (iii) ##s_n(x) = nsin(\frac{x}{n})## Homework Equations ##s_n(x) →...

Hello, In Calculus 2, sequences and series are introduced and do I have to say that most of the examples are trivial and even the exercises are either trivial or those that require experience. I hope someone can suggest a book where one can learn solving not-so-obvious series problems that...

Homework Statement what is the general formula for the sequence (1/1*3+1/3*6+1/6*10+1/10*15...) Homework Equations i used the equation n/mn+1 but am not able to use it for this sequence The Attempt at a Solution I found the sequence of the denominators which is (1/2)n^2+(1/2)n...

I have to prove that the cardinality of the set of infinite sequences of real numbers is equal to the cardinality of the set of real numbers. So: A := |\mathbb{R}^\mathbb{N}|=|\mathbb{R}| =: B My plan was to define 2 injective maps, 1 from A to B, and 1 from B to A. B <= A is trivial, just...

I'm trying to find a sequence that has subsequences that converge to every integer. The question before that was the same but just for the positive integers, for which i gave {1,1,2,1,2,3...} but I'm struggling to include the negatives. Thanks

Homework Statement For an increasing sequence of numbers, how many other sequences could this be the average sequence of. Homework Equations Where the average sequence, a[i] = 0.5( s[i] + s[i+1] ) The Attempt at a Solution If there's n terms in the original sequence. The number of...

Hi, True or False: Every infinite sequence of natural numbers, who's terms are randomly ordered, must contain every possible subsequence of any length, including infinity. For example, does the infinite and random sequence \small M of natural numbers require that the subsequence {59,1,6}...

After losing marks in an exam due to significant figures, I have decided to clear up all my doubts about this concept. But since my teacher hasn't been very helpful, I've decided to post my question here. I understand the rules for significant figures in both single-step...

Homework Statement If P r=(n-r)(n-r+1)(n-r+2)...(n-r+p-1) Qr= r(r+1)(r+2)...(r+q-1) Find P1Q1+P2Q2+... +Pn-1Qn-1 Homework Equations The Attempt at a Solution I tried to bring the general term in...

In this link: http://people.ischool.berkeley.edu/~johnsonb/Welcome_files/104/104hw3sum06.pdf For number 8.4... Why don't we just say... |s_n||t_n| < \frac{\epsilon}{M} M = \epsilon? Thanks in advance

Hi I don't understand the logic in the picture i added. They say that "that sum of the series = the limit of the sequence" The limit is 2/3 BUT the sum, Ʃ, must be 2*1/(3*1+5) + (2*2/(2*3+5) + 2*3/(2*3+5) ...+ Which is obviously much larger than 2/3 if all the terms are added together?? it's...

Homework Statement I'm having trouble with these here.. it's been a while since I've done sequences and I can't seem to make this work with Standard Limits equations. Clearly the answer given by Wolfram solver is there after the = but i'd like to know the reasoning behind it. Anyone that...

Homework Statement Read this passage and then answer the questions that follow We know that, if a_1,a_2,...,a_n are in Harmonic Progression, then \frac{1}{a_1},\frac{1}{a_2}...,\frac{1}{a_n}, are in Arithmetic Progression and vice versa. If a_1,a_2,...,a_n are in Arithmetic Progression with...

Homework Statement Find the sum of the sequence: 2, -2/3, 2/9, -2/27, 2/81, . . . Homework Equations The Attempt at a Solution I can see that the number is multiplied by -1/3, but I'm unsure of how to find the sum. Any pointers?

The effect of changing values in sequences?? I have been given a maths assignment and have been given equations \(u_{n+1}=2u_{n}+2\) and asked what is the effect if the value \(u_{0}\) is changed? I used multiple values both positive and negative and have only noticed taht when it is a high...

Homework Statement Let an be a bounded sequence and bn such that the limit bn as n→∞ is b and 0<bn ≤ 1/2 (bn-1) Prove that if: an+1 ≥ an - bn, then lim an n→∞ exists. Homework Equations The Attempt at a Solution as 0<bn ≤ 1/2 (bn-1) the sequence bn is...

Homework Statement Let an be a bounded sequence and bn such that the limit bn as n→∞ is b and 0<bn ≤ 1/2 (bn-1) Prove that if: an+1 ≥ an - bn, then lim an n→∞ Homework Equations The Attempt at a Solution no clue :(

Homework Statement Let f be a 2π-periodic function (can be any periodic really, not only 2π), and let g be a smooth function. Then lim_{n\rightarrow∞}\int^{B}_{A} f(nx)g(x) converges to \frac{1}{2π}\int^{2π}_{0}f(x) The Attempt at a Solution So far, I've come up with somewhat of...

Homework Statement Prove the following theorem, originally due to Cauchy. Suppose that (a_{n})\rightarrow a. Then the sequence (b_{n}) defined by b_{n}=\frac{(a_{1}+a_{2}+...+a_{n})}{n} is convergent and (b_{n})\rightarrow a. Homework Equations A sequence (a_{n}) has the Cauchy property...

Let x_n and y_n be two convergent sequences with different limits. Show that the set {x_n : n€N} n {y_n : n€N} is finite. Attempt: by definition, for each £>0 there exists an N such that |x_n - x|<£ and similarly |y_n - y|<£ holds for every n with n>N. Take £=(x-y)/3 and assume that x_n and...

Hello everyone! Let $a_n$ and $b_n$ be two sequences such that $a_n \leq b_n$ for all $n$. Let $A_n = \sup \{a_m \; | \; m \geq n\}$ and $B_n = \sup \{b_m \; | \; m \geq n\}$. I want to prove that $A_n\leq B_n$. I attempted a proof by contradiction: Assume $A_n > B_n$ for some $n$. If $A_n =...

I have a linear recurrence sequence and am having a problem understanding what to do when the ratio does not seem to be the same between each of the terms, so Terms; 4, 1.4, 2.44, 2.024... (n = 1,2,3...) How do I find a the ratio of these terms, and if there is none, please advise how I...

cn= (4n)/(n+4n^(1/n)) When i set it up i think i should use l'hopital but I am confused what to do with the 4n^(1/n) term. an=(7^(2n))/(n!) I know this is a geometric sequence and top and bottom increase initially then tend to 0, but I am lost on how to show the work. should i expand...

Homework Statement Assume that \{ a_n\}\rightarrow 0 . Use the definition of limit to prove that \{ a_n^2\} \rightarrow 0. Homework Equations Definition of limit. For all ε>0 there exists N s.t. n>N implies |a_n - L|<ε. The Attempt at a Solution I know why this is true... if the sequence...

Homework Statement I have to proof that the sequence (2^n +n^2)/(3^n + 5n^4) converges en calculate its limit using the sqeeuze theorem. Homework Equations (2^n +n^2)/(3^n + 5n^4) http://www.proofwiki.org/wiki/Squeeze_Theorem#Sequences Theorem 1: Let p\in2N en x\inR with |x|< 1. Then the...

Homework Statement Let (xn)n\inℕ and (yn)n\inℕ be Cauchy sequences of real numbers. Show, without using the Cauchy Criterion, that if zn=xn+yn, then (zn)n\inℕ is a Cauchy sequence of real numbers. Homework Equations The Attempt at a Solution Here's my attempt at a proof: Let...

Hi, I am wondering how one would go about an ε, N proof for a recursively defined sequence. Can anyone direct me to some reading or would like to provide insights of their own? This isn't for a homework problem... just general curiosity which I could not satisfy via search! Thank you.

Homework Statement I want to prove that if a sequence a[n] is cauchy then a[n] is a converging sequence Homework Equations What I know is: a[n] is bounded any subsequence is bounded there exists a monotone subsequence all monotone bounded sequences converge there exists a...

I am not getting anywhere with this problem. Prove the Schwarz's and the triangle inequalities for infinite sequences: If $$ \sum_{n = -\infty}^{\infty}|a_n|^2 < \infty\quad\text{and}\quad \sum_{n = -\infty}^{\infty}|b_n|^2 < \infty $$ then $\displaystyle\left(\sum_{n = -\infty}^{\infty}|a_n +...

Sorry for the rather vague title! Homework Statement Given: Two Banach spaces A and B, and a linear map T: A\rightarrow B The sequences (x^n_i) in A. For each fixed n, (x^n_i) \rightarrow 0 for i \rightarrow \infty. The sequences (Tx^n_i) in B. For each fixed n, (Tx^n_i) \rightarrow y_n...

I read the proof of the proposition "every cauchy sequence in a metric spaces is bounded" from http://www.proofwiki.org/wiki/Every_Cauchy_Sequence_is_Bounded I don't understand that how we can take m=N_{1} while m>N_{1} ? In fact i mean that in a metric space (A,d) can we say that...

If A and B are vector spaces over ℝ or ℂ show that a sequence (a_n, b_n) in A×B converges to (a,b) in A×B only if a_n converges to a in A and b_n converges to b in B as n tends to infinity. To me this statement sounds pretty intuitive but I have been having trouble actually proving it...

Homework Statement \stackrel{lim}{n\rightarrow \infty} (-1)^n \frac{n}{n + 1} Homework Equations The Attempt at a Solution The answer is that the limit oscillates between -1 and 1, but I was wondering if there was an analytic was of showing this.

Hello, I am curious to know that if we take some seqence, a_n, and take the limit as the the terms of the sequence goes to infinity, will the sequence head towards the same value that the the sum of the infinite amount of terms added together? (I hope I worded that properly...)

Homework Statement Prove that the given sequence diverges to infinity. {an} = (-n^4+n^3+n)/(2n+7) Homework Equations Diverges definition The Attempt at a Solution So far I have: Let M>0 and let N= something. I'm having a hard time figuring out what N should equal for the...

Homework Statement Determine whether the given limit exists and find their values. Give clear explanations using limit properties. Homework Equations lim n--->∞ (n^2)/n! The Attempt at a Solution I know that the limit is 0, but I don't know how to show it in detailed steps...

can anyone show me how to do this question ? thanks ... express (1+x^2)/((1+x)(1+2x)) in partial fraction. (this step i know the solution ) hence,find the constant term in the expansion if (1+x^2)/(-3x(1+x)(1+2x)) in ascending power of x .( then this one don't know ,please help me ) thanks ...

I was attempting to find a counterexample to the problem below. I think I may have, but was ultimately left with more questions than answers. Consider the space, L, of all bounded sequences with the metric \rho_1 \displaystyle \rho_1(x,y)=\sum\limits_{t=1}^{\infty}2^{-t}|x_t-y_t| Show that a...

Hi guys, I'm doing some exercises in which given a recursive sequence and its first term, I have to find the general formula/term. I am stuck in two and I would like some help. Thanks in advance. Now, the sequences: 1) a1=1, an+1= an + ((-1)^(n+1))n^2 So, the first terms are: a2=2...

Can someone guide me toward using my TI-89 Titanium calculator for sequences? I would like to be able to PUT IN a sequence of numbers and have it GIVE ME the formula. Not vice versa please. Thanks.

Hello. Having already learned about infinite series and sequences in my calculus class, I'm quite interested in them and especially in learning more about them. If any of you have in mind any good books on the subject which you can recommend to me, it will be very much appreciated...

Def: A low discrepancy sequence is a uniformly distributed sequence with minimal discrepancy, O(logN/N). Question: Let <x> denote the fractal part of an irrational number x. Let (<x_n>) be an arbitrary low discrepancy sequence. Is it always true that : \lim_{n \to +\infty}|<x_n - x_{n-1}> -...

Sequences and series help... [b]1. Homework Statement 3+3a+3a^2+...∞ is = to 45/8 where a>0,then a is...? [b]3. The Attempt at a Solution since it is a g.p so using S=(a(rn-1))/(r-1) for r>1 ive all the values except for "n"..can someone help...:/

Homework Statement Determine whether the series converges or diverges Sum from n=1 to infinity ((e^(1/n))/n) Homework Equations I am trying to use the limit comparison test to prove it. The Attempt at a Solution an = (e^(1/n))/n bn = e/n an/bn = e^(1/n)/e lim n->...