Sequences Definition and 590 Threads

Homework Statement (excuse lack of latex) show that if SUM(zn)= S and SUM(wn= T, then SUM(zn + wn) = S + THomework Equations The Attempt at a Solution so if I'm doing this right, this is pretty easy, i think. they want me to use a theorem that says if zn=xn +iyn, and SUM(zn)= S, where S = X +...

Homework Statement Let (u_{n})_{n} be a real sequence such that lim u_{n} = 0 as x→∞ and let (v_{n})_{n} be a bounded sequence. Show that lim (u_{n})_{n}(v_{n})_{n} = 0 as x→∞ Homework Equations The Attempt at a Solution Since (v_{n})_{n} is bounded then it has a least upper bound and...

Let A be a non-empty subset of the complex plane and let b ∈ ℂ be an arbitrary point not in A. Now define d(A,b) := inf{|z-b| : z ∈ A}. Show that if A is closed, then there is an a ∈ A such that d(A,b) = |a-b|. Ok so basically what I did was begin by choosing some arbitrary element of A and...

I know that Q (rational numbers) are countable and that the finite cartesian of Q with itself, Q^n is countable but is it true that the countably infinite cartesian product of Q with itself is countable? The set of all rational sequences are isomorphic to Q^∞ (here I am saying Q^∞ is the...

Homework Statement I'm just trying to find liminf and limsup for: Homework Equations E_n = (n, n+2) The Attempt at a Solution Since every subset occurs a finite number of times, would I say that limsup is the empty set? Is being bounded by nothing the same thing as being bounded by...

Homework Statement a) Prove that \ell_\infty \mathbb({R}) is a subspace of \ell \mathbb({R}) b) Show that \left \| \right \|_\infty is a norm on \ell_\infty (\mathbb{R}) The Attempt at a Solution For a) I guess we have to show that \vec{x} + \vec{y} \in \ell_\infty \mathbb({R})...

Homework Statement This is a homework question for a introductory course in analysis. given that a) the partial sums of f_n are uniformly bounded, b) g_1 \geq g_2 \geq ... \geq 0, c) g_n \rightarrow 0 uniformly, prove that \sum_{n=1}^{\infty} f_n g_n converges uniformly (the whole...

Homework Statement Suppose {Xn}, {Yn} are sequences in ℝ and that |Xn-Yn|→0. Show that either: a) {Xn} and {Yn} are both divergent or b) {Xn} and {Yn} have the same limit. Homework Equations N/A The Attempt at a Solution I first prove that lim(Xn-Yn)=lim(Xn)-lim(Yn). I am not...

Homework Statement Prove if sequence a_{n} converges and sequence b_{n} diverges, then the sequence a_{n}+b_{n} also diverges. Homework Equations The Attempt at a Solution My professor recommended a proof by contradiction. That is, suppose a_{n}+b_{n} does converge. Then, for...

Homework Statement a_n=(\frac{1}{n})^{\frac{1}{\ln{n}}} Homework Equations The Attempt at a Solution \lim_{x\rightarrow \infty}(\frac{1}{n})^{\frac{1}{\ln{n}}} y=(\frac{1}{n})^{\frac{1}{\ln{n}}} \ln{y}=\frac{\ln{\frac{1}{n}}}{\ln{n}} \ln{y}=\frac{\ln{1}-\ln{n}}{\ln{n}}...

Homework Statement The problem provides a sequence. Let x(n) ={1,-2,4,6,-5,8,10} where x(-4)=1, x(-3)=-2,etc. Generate x1(n) = 3x(n+2)+x(n+4)-2x(n) The Attempt at a Solution x = [0, 0, 1, -2, 4, 6, -5, 8, 10, 0, 0, 0, 0]; n = -6:6; x1 = x * 3; n1 = n-2; x2 = x * 2; n2 = n; x3 = x; n3 =...

Homework Statement If I have a sequence {Pn} and I know that lim Pn = p, can I call {Pn} infinite? I am trying to use this result in a real analysis proof. I know B(p; r) intersection S is non-empty and I need to show that it has indefinitely many points. I can show that {Pn} is a subset of...

Hi, I'm studying infinite series and was wondering if someone could recommend me a gigantic list of examples of series and proofs of weather they converge or not.

Homework Statement let (An) be a sequence in R with |summation from n=1 to infinity(An)|< infinity. Prove lim as n goes to infinity of ((A1 +2A2+...+nAn)/n) = 0 Homework Equations The Attempt at a Solution I think |summation from n=1 to infinity(An)|< infinity means the summation...

My question involves supremums and their implications: say I have the sequences \left\{x_{k}\right\}_{k=1}^{\infty} and \left\{y_{k}\right\}_{k=1}^{\infty} and I know sup \left\{x_{k}:k\in N \right\} \leq sup \left\{y_{k}:k\in N \right\} What can I say about the sequence...

Homework Statement Q.: Show that if log a, log b and log c are three consecutive terms of an arithmetic sequence, then a, b and c are in geomtric sequence. Homework Equations Un = a + (n - 1)d and Sn = \frac{a(r^n - 1)}{r - 1} The Attempt at a Solution Attempt: Consider...

Homework Statement Q.: The sum of the first five terms of a geometric series is 5 and the sum of the next five terms is 1215. Find the common ratio of this series. Homework Equations Sn = \frac{a(r^n - 1)}{r - 1} The Attempt at a Solution a + ar + ar^2 + ar^3 + ar^4 = 5 ar^5 +...

Suppose we're in a general normed space, and we're considering a sequence \{x_n\} which is bounded in norm: \|x_n\| \leq M for some M > 0. Do we know that \{x_n\} has a convergent subsequence? Why or why not? I know this is true in \mathbb R^n, but is it true in an arbitrary normed space? In...

Homework Statement Theorem 1.4: Show that every Cauchy sequence is bounded. Homework Equations Theorem 1.2: If a_n is a convergent sequence, then a_n is bounded. Theorem 1.3: a_n is a Cauchy sequence \iff a_n is a convergent sequence. The Attempt at a Solution By Theorem 1.3, a...

Homework Statement I wrote a test and the question was something like this 2, 4, 6... 108 It said... " Find tn" Does this just mean find any term number that isn't given? I just plugged in t5 for the arithamtic logic and solved.. don't know if it was right, does anyone know...

Homework Statement Determine the values of x for which the function, for n>=1, is increasing, decreasing, bounded below or bounded above. The function is (x^n)/n Homework Equations The Attempt at a Solution I thought about taking the derivative of the function, and setting it to 0. To find...

Anyone have a good example of a closed subset of a topological space that isn't closed under limits of sequences?

Hi there, I came across the following problem and I hope somebody can help me: I have some complete metric space (X,d) (non-compact) and its product with the reals (R\times X, D) with the metric D just being D((t,x),(s,y))=|s-t|+d(x,y) for x,y\in X; s,t\in R. Then I have some sequences...

Homework Statement Just a quick question I was looking to have cross checked… Q. Find un, the nth term of sequence -5, 0, 5, 10,… Homework Equations un = a + (n-1)d The Attempt at a Solution -5 + (n-1)5 -5 + 5n - 5 5n-10 The answer in the book...

So if you have a countably infinite set \{ x_n \} and consider also the sequence (x_n), what's the relationship between \sup \{ x_n \} and \limsup x_n?

Homework Statement Let an be monotone sequences. Prove or give a counterexample: The sequence cn given by cn=k*an is monotone for any Real number k. The sequence (cn) given by cn=(an/bn) is monotone. Homework Equations The Attempt at a Solution On the first one, I don't...

If \{a_n\}\to A, \ \{a_nb_n\} converge, and A\neq 0, then prove \{b_n\} converges. Let \epsilon>0. Then \exists N_1,N_2\in\mathbb{N}, \ n\geq N_1,N_2 |a_n-A|<\frac{\epsilon}{2} And let \{a_nb_n\}\to AB So, |a_nb_n-AB|<\epsilon I don't know how to show b_n is < epsilon.

Hope someone could give me some help with a couple of problems. First: Proof of - A function f:G -->Complex Plane is continuous on G iff for every sequence C(going from 1 to infinity) of complex numbers in G that has a limit in G we have limit as n --> infinity f(C) = f(limit as n...

The population of a country is 15.2 million and is growing at a steady rate of 2.7% annually. a) What was the population one year ago b) What was the population four years ago So I did this: Un+1 = Un-1 * ((n-1)*0.973) Doing so, I got for the first answer 14,789,600 but the book got...

I know that there are particular formulas for finding the geometric/arithmetic/ and recursive sequences or series with \Sigma. But is there a general formula for finding the sum for all three types? For example, what if I was asked to find a sum of a particular finite sequence but I don't know...

There's a game that's been around for a long time called Connect 4. It is a 2-player game consisting of 7 columns that can hold 6 discs each. The players alternate dropping a disc of their color into one of the 7 columns until a player has 4 in a row, either horizontally, vertically, or in a...

Homework Statement Let a0, a1, a2..., be defined by the formula an = 3n + 1, for all integers n >= 0. Show that this sequence satisfies the recurrence relation ak = ak-1 + 3, for all integers k >=1. Homework Equations for all integers n >= 0, an = 3n + 1 for all integers k...

I have been asked to find if the following sequence is increasing or decreasing: an = ne^-n So I first multiplied thru by e^n to get: n/e^n Then, I did n+1, so (n+1)*e^-(n+1). I moved the negative exponent to the bottom to get (n+1)/(e^(n+1)) I guess my first question is did I start...

Homework Statement Determine whether the sequence with the given nth term is monotonic. Find the boundedness of the sequence. a_n = ne^{-n/2} Homework Equations I don't know The Attempt at a Solution I have absolutely no idea what a monotonic sequence is or how to find the...

Homework Statement For every epsilon > 0, there exists an N\in N such that, for every j >= N, |f(i,n) - g(n)|<epsilon for every n\in N. In addition, for every fixed j\in N, (f(i,n)) converges. Prove that (g(n)) converges. Homework Equations f: N x N --> R, g: N --> R The Attempt at...

! Sequences and series "limit" question, is my solution correct? Homework Statement [PLAIN]http://img233.imageshack.us/img233/7195/sands2010q1.gif Homework Equations The Attempt at a Solution Solution posted in image above, want to know if its correct

Consider two sequences, {a_n} and {b_n}. If there is a one-to-one correspondence between these sets, can we conclude anything about their behavior considering, say, that we know that one is convergent? Going further, can we conclude anything about the series resulting from these sequences?

Homework Statement For each of the following sequences (fn), find the function f such that fn --> f. Also state whether the convergence is uniform or not and give a reason for your answer. Homework Equations a.) fn(x) = 1/xn for x greater than or equal to 1 b.) f[SUB]n[SUB](x) =...

So I had this question in PF chat, but I decided this would be a better place for it. Say I have two sets, S and S'. S is the set of all convergent sequences. S' is the set of all convergent series...es. Is S larger than S', and if so, how much larger?

Homework Statement For a sequence a_n: If lim (a_n) =2, use the definition of a limit to show that lim (a_n / n) = 0 all limits are as n goes to infinity The Attempt at a Solution I know that I need to show: Give any \epsilon>0 there is some M so that if n>M then |a_n / n| <...

Given the sequence: if n=1, an = 2 if n>1, an+1 = 1/2(an + 3/an) prove that this sequence is decreasing im having trouble with recursively defined sequences. I know I am supposed to use induction in some way, but its not that straitforward with the 'double sequence' in the an+1...

ok so, a) If s sub n→0, then for every ε>0 there exists N∈ℝ such that n>N implies s sub n<ε. This a true or false problem. Now this looks like a basic definition of a limit because s sub n -0=s sub n which is less than epsilon. n is in the natural numbers. But, I thought there should be...

"Cauchy" Sequences in General Topological Spaces Is there an equivalent of a Cauchy sequence in a general topological space? Most definitions I have seen of "sequence" in general topological spaces assume the sequence converges within the space, and say a sequence converges if for every...

Homework Statement So I was helping my roommate with his homework, and it has the following problem: Homework Equations The Attempt at a Solution We tried a Fibonnaci-type sequence, but that really didn't work. And we don't know any other types of sequences. Should I try some...

Homework Statement I'm approaching this problem from a different method than conventially shown. Homework Equations if lim=infinity for all M>0, there exists a N such that n>N => {s(n)}>=M The Attempt at a Solution this can be rewritten as: {s(n)} is a sequence. If...

I have 2 questions. How do you use differetiation to prove whether sequence is monotonic? For example: 1/n+ln(n) My 2nd question is, how do you prove whether sequence is EVENTUALLY monotonic?

Homework Statement If f:S->Rm is uniformly continuous on S, and {xk} is Cauchy in S show that {f(xk)} is also cauchy. Homework Equations The Attempt at a Solution Since f is uniformly continuous, \forall\epsilon>0, \exists\delta>0: \forallx, y ∈ S, |x-y| < \delta =>...

Homework Statement For the following series, write formulas for the sequences an, Sn and Rn, and find the limits of the sequences as n-->infinity Homework Equations N/AThe Attempt at a Solution an is easy, = the limit of which does not exist. This is where I get stuck, I know Sn= But I don't...

Homework Statement This is a nice problem, compared to the previous one, at least it seems so. One needs to show that a metric space (X, d) is complete iff for every nested sequence ... \subseteqA2\subseteqA1 of nonempty closed subsets of X such that diam An --> 0, the intersection of the...

Homework Statement A) In a certain geometric sequence every term is the sum of the two preceding terms, viz. the Fibonacci sequence, what can be said about the common ratio of the sequence? So how do I go from 1,1,2,3,5,8,13,21,34... to (1+/-sqrt(5))/2? Then find numbers A and B such (for...