Sequence Definition and 1000 Threads

Def. Let $\{z_j\}$ be a sequence of non-zero complex numbers. We call the exponent of convergence of the sequence the positive number $b$, if it exists, $$b=inf\{\rho >0 :\sum_{j=1}^{+\infty}\frac{1}{|z_j|^{\rho}}<\infty \}$$ Now consider the function $$f(z)=e^{e^z}-1$$ Find the zeros $\{z_j\}$...

Good Day, My friends and I are stuck on solving the last part of the attached problem. The solution is 2^[(n^2 + n)/2] - 1. Can anyone help us with solving this? Thanks & Regards, Nicodemus

A few years ago I made an exploration of some simple concepts that led to some fun facts and a sequence of numbers I later learned is quite well-known. It may even rival the Fibonacci sequence in its fame. I thought it might be nice to collect my notes and present my findings here. I thought I...

I have a non-increasing sequence of random variables \{Y_n\} which is bounded below by a constant c, \forall \omega \in \Omega. i.e \forall \omega \in \Omega, Y_n \geq c, \forall n. Is it true that the sequence will converge to c almost surely? Thanks

Hi All, I posted this text below in another forum, but didn't get much feedback. I realize I might sound extremely ignorant, but I need directions. Thanks. "Awhile ago I decided I wanted to learn Relativity comprehensively. I was told that I better brush up on my Differential Geometry...

\text{We need to prove that the sequence} \ a_{n} = \{n^{2}/2^{n}\} \ \text{converges to 0} \\ \text{Consider the sequence {n/ 2n} = { 0, 1/2, 1/2, 3/8, 1/4, 5/32, ...}. The terms get smaller and smaller.}\\ \\ \text{we can easily show that} \ n/2^{n}<=1/n \ \forall n>3 \\ \text{from the fact...

Homework Statement fn is a sequence of functions and sn is a sequence of reals such that 0 ≤ fn(x) ≤ sn for all x. I want to show that if \sum_{k=0}^{n}s_k is Cauchy then \sum_{k=0}^{n}f_k is uniformly Cauchy and that if \sum_{k=0}^{\infty}s_k converges then \sum_{k=0}^{\infty}f_k converges...

Homework Statement let ##a_n## be ##a_{n+1}=\frac{1}{4-3a_n} \quad n≥1## for which values of ##a_1## does the sequence converge? which is the limit? The Attempt at a Solution ##0<a_1<\frac{4}{3}## because if ##a_1>\frac{4}{3}→a_2<0## not possible. Now let's assume ##a_n## converges to M. I...

Hi PF, I have a device (with a microcontroller) which generates random numbers. when I analyzed those numbers using Matlab software I found that it is following a uniform distribution. How can I mathematically (any algorithm?) convert this random output to a gaussian one. Also I would...

Could you check my solution please? Homework Statement find out for which values of ##\lambda>0## the sequence ##(a_n)## ,defined by ##a_1 = \frac{1}{2}, \quad\quad a_{n+1} = \frac{1}{2} (\lambda +a_n)^2, \quad n\in \mathbb{N^*}## converges. If ##(a_n)## converges, find the limit. The...

I think the solution I've found makes sense, but I'd like it to be double-checked. Homework Statement Let ##(a_n)## be a limited sequence and ##(b_n)## such that ##0≤b_n≤ \frac{1}{2} B_{n-1} ## Prove that if ##a_{n+1} \ge a_{n} -b_{n}## Then ##\lim_{n\to \infty}a_n## exists...

Hi, In Baby Rudin, Thm 3.6 states that If p(n) is a sequence in a compact metric space X, then some subsequence of p(n) converges to a point in X. Why is it not the case that every subsequence of p(n) converges to a point in X? I would think a compact set would contain every sequence...

Intersection of a sequence of intervals equals a point (Analysis) Homework Statement Let A_{n} = [a_{n}, b_{n}] be a sequence of intervals s.t. A_{n}>A_{n+1} and |b_{n}-a_{n}|\rightarrow0. Then \cap^{∞}_{n=1}A_{n}={p} for some p\inR. Homework Equations Monotonic Convergent Theorem If...

Homework Statement Given is a sequence: sin(1), cos(sin(1)), sin(cos(sin(1))) etc. Find the limit of the sequence or prove it diverges.Homework Equations ? The Attempt at a Solution One way to prove a sequence diverges is to find two subsequences which converge to different limits, but I...

Homework Statement I'm looking for a sequence of simple functions fn that converges uniformly to f(x)=x2 on the interval [0,1]. Homework Equations I know a simple function is one that can be written as \sum^{n}_{k=1}a_{k}1_{D_{k}}(x) where {D1,...,Dn} is collection of measurable sets...

Hi All, I have an unknown 16S ribosomal RNA gene sequence below that I am trying to identify. I know its a bacteria but anyone know of a site/program online that allows you to enter the sequence below and it gives the likely close match? TAGGGAATCTTCCGCAATGGACGAAAGTCTGACGGA...

I'm having issues doing a sequence diagram for a program. Like this format here: http://upload.wikimedia.org/wikipedia/commons/thumb/9/9b/CheckEmail.svg/220px-CheckEmail.svg.png The scenario is the following: Placing a bet - a customer places a bet for a game. The customer has to be...

Homework Statement Determine the limit of the sequence: an = (1+(5/n))2n Homework Equations L'hopitals rule, or at least that's what I'm thinking. Otherwise, general formulas for determining the limit of a sequence. The Attempt at a Solution an = (1+(5/n))2n Considering the...

What does the "N" mean in a Cauchy sequence definition? Hi everyone, I have a question regarding Cauchy sequences. I am trying to teach myself real analysis and would appreciate any clarification anyone has regarding my question. I believe I have an intuitive understanding of what a Cauchy...

Homework Statement Find a formula that generates the sequence: 2/(3 x 4), -3/(4 x 5), 4/(5 x 6), -5/(6 x 7), . . . Homework Equations The Attempt at a Solution Here is what I have so far: a_n = -1(n/((n + 1)(n + 2))) Now, I'm stuck. The formula generates a negative number every time. I...

Homework Statement In an arithmetic sequence, the 11th term is 53 and the sum ofof the 5th and 7th terms is 56. Find the first 3 terms of the sequence. Homework Equations The Attempt at a Solution I'm trying to use the formula: tn= t1+(n-1)d but don't have right numbers. please...

If {x} is a sequence of rationals, I understand every real number will be a limit point. However, sequences have an order to them, right? So if this sequence of all rationals is monotonically increasing, then it will converge to infinite and all subsequences will have to converge to infinite. If...

Homework Statement Find the limit of the sequence given by S_{n}=\frac{n^{n}}{n!} Homework Equations lim_{n->∞}\frac{n^{n}}{n!} The Attempt at a Solution I know the sequence diverges, but that doesn't mean the limit is also ∞, right?

Homework Statement How many terms are in each sequence? 12, 4, 4/3, ..., 4/729 Homework Equations The Attempt at a Solution using tn=t1(r)(n-1) ? I am lost

Homework Statement Homework Equations The Attempt at a Solution This is what I have so far: x_{n+1}=\frac{x^5_n + 1}{5x_n}=1 x_{n+2}=\frac{x^5_{n+1} + 1}{5x_{n+1}}=\frac{1^5+1}{5} I think you have to do some sort of repeated substitution but I don't quite see it. Any help? Thanks.

Homework Statement Let {pn}n\inP be a sequence such that pn is the decimal expansion of \sqrt{2} truncated after the nth decimal place. a) When we're working in the rationals is the sequence convergent and is it a Cauchy sequence? b) When we're working in the reals is the sequence...

I have across the following argument, which seems wrong to me, in a larger proof (Theorem 4 on page 9 of the document available at http://www.whitman.edu/mathematics/SeniorProjectArchive/2011/SeniorProject_JonathanWells.pdf). I would appreciate if someone can shed light on why this is true...

Hi, this was a midterm problem in a probability class. Homework Statement A fair coin is tossed repeatedly with results Y(0), Y(1)... that are 1 or 0 for heads or tails. For n>0, define a new sequence X(n) = Y(n)+Y(n-1), i.e. the number of 1's in the last two tosses. Is Xn a markov...

Homework Statement Show that the following sequence \sum\limits_{n=1}^\infty \frac{1}{n^\alpha} for all real \alpha > 1 converges and for all real \alpha \leq 1 diverges. The Attempt at a Solution All I know is that the Abel-Summation is the only useful thing here, but I got no clue...

For any real ##x > 0##, prove that the sequence ##n^{n^{-x}}## is bounded (and if possible, monotonically decreasing after some point). The catch is that logarithms and the exponential constant cannot be used. We must arrive at the proof using fairly "primitive tools" If you look at the graph...

Homework Statement Does the following sequence converge, or diverge? a_{n} = sin(2πn) Homework Equations The Attempt at a Solution \lim_{n→∞} sin(2πn) does not exist, therefore the sequence should diverge? But it actually converges to 0? I appreciate all help thanks. BiP

I'm trying to understand divergence of a sequence (not series). What methods can I use to prove divergence? I know that convergence can be proven using various methods, such as squeeze theorem and sum, difference, product and quotient rule etc. Could I use the following to prove divergence...

Homework Statement Prove that, s^{*} = \lim_{n \rightarrow \infty} \sup_{k \geq n} s_k Assume that s^{*} is finite. Homework Equations Definition of s^{*} is here: http://i.imgur.com/AWfOW.png The Attempt at a Solution I started out writing what I know. By assuming s^{*} is...

Hi, I'm sure x_t is a decreasing sequence while y_t is an increasing one. It feels like it should be simple to prove, but I just can't do it. Any suggestions would be great! Thanks, Peter x_t and y_t are defined iteratively by two equations: 1. y_(t+1) = bq x_t + b(1 - q) y _t 2...

Sorry to spam my problems all over this forum but series have me struggling somewhat. Last problem on my homework is the sequence an defined recursively by: a1=1 and an+1= \(\frac{a_n}{2}\) + \(\frac{1}{a_n}\) First part was the only part i know how to do. it was to find an for n=1 through 5...

Does anybody know if this statement is true? \sum fn converges absolutely and uniformly on S if ( fn) converges uniformly. Also if R is the radius of convergence and |x|< R does this imply uniform and absolute convergence or just absolute convergence.

Homework Statement Show that the if lim bn = b exists that limsup bn=b. The Attempt at a Solution Let limsup = L and lim = b We know for all n sufficiently large |bn-b|<ε |bn| < b+ε Therefore L ≤ b+ε and |bn| < L ≤ b+ε I'm trying to get |bn-L|<ε or |L-b|<ε both of which I...

Hi. I'm currently tutoring this student with High school math, and I'm completely stumped on this question that he was asked on his test. I'm hoping the community can help me help my student! Homework Statement The student was presented with two sums of a geometric sequence (eg, Sum of...

Hi, I am unsuccessful at showing that the sequence √n + 1/n is an ascending monotone one, i.e. that a_n+1 > a_n for any n, greater than 2 let's say. I have proven that it is not bounded from above and is bounded from below. Any ideas, suggestions, please?

Homework Statement Be K \geq 1. Conclude out of the statement that \lim_{n \to \infty } \sqrt[n]{n} = 1, dass \sqrt[n]{K} = 1 The Attempt at a Solution \lim_{n \to \infty } \sqrt[n]{K} \Rightarrow 1 \leq \sqrt[n]{K} \geq 1 + ... I got issues with the right inequality...

Hello, My instructor, whilst trying to prove that liminf of sequence a_n = limsup of sequence a_n = A, _ wrote that since we know that a_n0-ε<an<a_n0+ε → a_n0-ε ≤ A ≤ A ≤ a_n0+ε...

Homework Statement Let d0, d1, d2,... be defined by the formula dn = 3n - 2n for all integers n ≥ 0. Show that this sequence satisfies the recurrence relation. dk = 5dk-1 - 6dk-2.Homework Equations The Attempt at a Solution I found that dk = 3k - 2k dk-1 = 3k-1 - 2k-1 dk-2 = 3k-2 - 2k-2...

Homework Statement Study the convergence of the following sequences a_{n} = \int^{1}_{0} \frac{x^{n}}{1+x^{2}} b_{n} = \int^{B}_{A} sin(nx)f(x) dx The Attempt at a Solution For the first one, I said it was convergent. I'm not exactly sure why though, my reasoning was...

Hello everyone! I am told that the limit of $\frac{x_{n+1}}{x_n}$ is $L>1$. I am asked to show that $\{x_n\}$ is not bounded and hence not convergent. This is what I got so far: Fix $\epsilon > 0$, $\exists n_0 \in N$ s.t. $\forall n > n_0$, we have $|\frac{x_{n+1}}{x_n}-L|<\epsilon$...

We know that for k\in\mathbb{N} we have, if: \displaystyle\lim_{n\to\infty}\left(a_{n}-a_{n-k}\right)=k\cdot a then: \displaystyle\lim_{n\to\infty}\frac{a_n}{n}=a When the reverse impliaction is also true? What do we have to assume to achieve if and only if theorem? I'm especially interested...

Homework Statement x_{n+1} = (x_{n} + 2)/(x_{n}+3), x_{0}= 3/4Homework Equations The Attempt at a Solution I've worked out a few of the numbers and got 3/4, 11/15, 41/56, 153/209, ... It seems to be monotone and bounded below indicating it does converge I think. I need help figuring out what...

Homework Statement Show that an increasing sequence is convergent if it has a convergent subsequence. The Attempt at a Solution Suppose xjn is a subsequence of xn and xjn→x. Therefore \existsN such that jn>N implies |xjn-x|<\epsilon It follows that n>jn>N implies |xn-x|<\epsilon...

Homework Statement Let {fn} be a sequence of measurable functions defined on a measurable set E. Define E0 to be the set of points x in E at which {fn(x)} converges. Is the set E0 measurable? Homework Equations Proposition 2: Let the function f be defined on a measurable set E...

Homework Statement Let's say that I have a set called M, which is a subset of real numbers. Let's say that I want to create a sequence {s_1, s_2, ..., s_3} with all of the members of M, which would be ordered in an ascending (increasing) order. For example, if M = {4, 5, 1, 3, 2}, then s_0 = 1...

Homework Statement Let b1=1 and bn=1+[1/(1+bn-1)] for all n≥2. Note that bn≥1 for all n in N (set of all positive integers). The Attempt at a Solution Prove that (b2k-1)k in N By definition, a sequence (an) is increasing if an≤an+1 for all n in N. SO, for this problem, must...