Sequence Definition and 1000 Threads

Hi guys, I am trying to find the z-tranform of the following equation: x(n) =-2n u(-n-1) Using the Z-transform definition,summation and geometric series I am getting 1/(1-2z-1) But according to my lecturer the answer is suppose to be 0.5/(1-2z-1) I have tried simplification/factorization and...

##a_1=1## ##a_{n+1}=\frac{1}{2} ( a_{n} + \frac{b}{a_n} )## This should converge to ##\sqrt{b}## but I seem not to be able to prove this. Could someone give me a hint.

Homework Statement Find the limit of the given sequence as n →∞ Homework Equations (1+n^2)^(1/ln(n)) The Attempt at a Solution Wolfram said the answer was e^2, though i cannot figure out why. Any help would be greatly appreciated.

Homework Statement Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... By considering the terms in the Fibonacci sequence whose values do not exceed four million...

Homework Statement Suppose x-2,x and x+6, where x is an integer, are consecutive terms in a geometric sequence S Determine x Homework Equations r=x/(x-2)=(x+6)/x The Attempt at a Solution x/(x-2)=(x+6)/x cross multiply x(x)=(x-2)(x+6) x^2=x^2+4x-12 x^2-x^2+4x-12/4=0 4x=12 x=3

Homework Statement Determine whether the sequence zn = n/((1+i)n) converges and rigorously justify your answer. Homework EquationsThe Attempt at a Solution I have attempted an ε-n proof using my limit as 0 (as exponentials grow faster than polynomials I assumed this was the correct limit)...

Homework Statement If a,b,c,d ∈R+ and a,b,c,d are in H.P. Then 1. a+ d > b+ c 2. a+ c > b+ d 3. a + b > c+d 4. a-b > c-d Homework Equations Don't know which equation to apply The Attempt at a Solution 1/a, 1/b , 1/c and 1/d would be in A. P 1/b -1/a = 1/d - 1/c

Homework Statement A sequence (an) of real numbers is defined by a(1) = 6, a(n) = (1/4)(2*a(n−1) − 3) for n >= 2. where n is a natural number show the sequence is monotonically decreasing Homework Equations None The Attempt at a Solution I tried to prove it by induction. Let P(n) be the...

Let ${a_n}$ and ${b_n}$ be two convergent sequences such that $\lim_{{n}\to{\infty}}|a_n-b_n|=0$. Give an $\epsilon-N$ proof to show that $\lim_{{n}\to{\infty}}a_n=\lim_{{n}\to{\infty}}b_n$. Right now, I've just said without loss of generality, assume that $a_n>b_n$ for all $n>N$, then since...

Homework Statement Determine whether the sequence is convergent or divergent Homework Equations an = {nn / n! } The Attempt at a Solution an = ( n . n. n ... n ) / 1 . 2 . 3 ... n ) ⇒ an =n [ ( n . n. n ... n ) / 1 . 2 . 3 ... n ) ] ⇒ as n → ∞ , an → ∞ This is further confirmed by...

Derive an exact formula (solve the recurrence definition) for the following recursive sequence: $s_n = 2_{s_n-1} - s_{n-2}$ where $n \ge 2$, and $s_0 = 4$, $s_1 = 1$. So I saw someone solving this by making it a differential equation or something? How would you do that? should I do let...

Homework Statement Determine whether the sequence converges or diverges. If it converges, find the limit. Here's the sequence: http://www4a.wolframalpha.com/Calculate/MSP/MSP89541ea2ag9dg617bcd6000050d52e94i67ei593?MSPStoreType=image/gif&s=39&w=66.&h=44. Homework Equations N/A The Attempt at...

Hi, I'm trying to solve this exercise but I really don't know how 1. Homework Statement Let X1, X2,.. be a sequence of iid random variables following a uniform distribution on (0,1). Define the random variable N≥2 as the first point in which the sequence (X1,X2,...) stops decreasing. i.e If...

The following question came up when me and a friend of mine were discussing some basic things about probability:Let $p$ be a real number in $[0,1]$. Does there exist a sequence $(x_1, x_2, x_3, \ldots)$ with each $x_i$ being either $0$ or $1$, such that $$ \lim_{n\to \infty} \frac{f(n)}{n} =p...

And what will be the peptide sequence formed? So the sequence is: ADN 3’- TAC-CGA-TAT-AGC-CGT-ATT -5’ so my guess is that it's AUG-GCU-AUA-UCG-GCA-UAA and the peptide sequence is: Start codon, alanin, isoleucine, serine, alanin, stop codon. My question is: What do those 3's and 5's mean...

How do i go about deriving a general eqn for the nth term of a sequence provided an eqn of the sum to the nth term is given in terms of n?

Homework Statement 5) Examine the following sequence: 5' GAATTC 3' 3' CTTAAG 5' If this sequence was disgested with an enzyme (ex. ECOri) which cut between the guanine and adenine bases on which would it have? A) 3' overhang B) Blunt ends C) 5' overhang D) Any of these Homework Equations n/a...

Homework Statement \ell is the set of sequences of real numbers where only a finite number of terms is non-zero, and the distance metric is d(x,y) = sup|x_n - y_n|, for all n in naural-numbers then the sequence u_k = {1,\frac{1}{2},\frac{1}{3},...,\frac{1}{k}, 0,0,0...} and...

I need to construct a cauchy sequence ${r}_{n}$ such that its rational for all n belonging to set of natural numbers, but the limit of the sequence is not rational when n tends to infinity. I know that all convergent sequences are cauchy but I can't randomly conjure up a sequence that satisfies...

What are the rules if you have a sequence f_n of real-valued functions on \mathbb R and consider the sequence f_n(x_n), where x_n is some sequence of real numbers that converges: x_n \to x. All I have found is an exercise in Baby Rudin that says that if f_n \to f uniformly on E, then f_n(x_n)...

Hello, Solving last exam and stuck in this exercise Homework Statement Consider an increasing sequence {xn} . We suppose ∃ x∈ℝ and {xnk} a sebsequence of {xn} and xnk→x a/ Show that for any n∈ℕ , ∃ k∈ℕ as n≤nk b/ Show that xn→x Homework Equations 3. The Attempt at a Solution [/B] For b/ it...

I have a sequence of natural numbers which is generated by a really complicated function. This function only takes in natural numbers. How do I solve for a function that is asymptotic to the sequence? If I have a sequence of ans, I want a function that satisfies f(n) ~ an as n -> infinity.

Homework Statement My professor wants us to program on Python, where we have a certain sequence, for example: sequence = ("one", "two", "three", "four") I need to replace one of the sequence elements (example: "one") with another element (example: instead of the word "one", I need to put...

Assume that I have absolutely no clue to what is the formula used to generate a sequence. How do I know what kind of formula that is? (Exponential / Linear / Polynomial / etc) Also assume that there is only 1 formula that generates the sequence. I have read somewhere before that: f'(x) ~...

First, I would like to clear up notation and the definition for sequences. What exactly is a sequence? I read somewhere that it is defined as a function ##f: \mathbb{N} \to \mathbb{R}##. But if this is the case, why do we only define functions based on the range of the function, e.g., ##\left \{...

Homework Statement A strand of length L begins life as all A's. Assume that each letter evolves independent of all the rest until today, 1000 generations later. Within each generation there is a ##\mu## probability that the letter mutates to either C, G, T. Finally, assume that once a letter...

Homework Statement I have a DNA sequence generated by L throws of a 4 faced die with probabilities ##\pi_A, \pi_C, \pi_G, \pi_T##. Each probability is unknown. Task: estimate the probability of each side of the die. Hint: use a random variable defined by the sequence that has a binomial...

Homework Statement I have a 4-letter DNA sequence (AGGA) that appears 10 times in a strand that is 1027 letters long. The probability of finding this sequence at any random position is 1/256. What is the Z-score of this observation? Homework Equations ##Z=\frac{(X-E(X))}{\sqrt{Var(X)}}## The...

Homework Statement tn=1+1/2+1/3+...+1/n - ln(n) a.) Interpret tn - tn+1= [ln(n+1)-ln(n)] - 1/(n+1) as a difference of areas to show that tn - tn+1 > 0. Homework EquationsThe Attempt at a Solution I have not started working on part b) yet, because so far I am stuck on part a). I just simplified...

Homework Statement Consider the sequence in the form of groups (1), (2,2), (3,3,3), (4,4,4,4), (5,5,5,5,5),... i. The 2000th term of the sequence is not divisible by (A) 3 (B) 9 (C) 7 (D) None of these ii. The sum of the first 2000 terms is (A) 84336 (B) 96324...

I am reading Walter Rudin's book, Principles of Mathematical Analysis. Currently I am studying Chapter 2:"Basic Topology". Although I can basically follow it, I am concerned that I do not fully understand the proof of Theorem 2.38. Rudin, Theorem 2.38 reads as...

I wonder. If I received the data (I don't know in what format) of all or part of my genome sequence, whould all the genes and SNP's that are physically in the same chromosome (not its homologous one) show together in the data or the process of sequencing and assembly mixes them up? I understand...

Hi , Everyone… one very general question regarding ic engine is suppose our vehical (Ic engine) Is at rest ,and we are ready to start vehical , when we kicked the kick By our legs then there is compression stroke..,right?, then when air fuel mixture is sucked in cylinder ( When there is...

Find the nth term of the sequence; ## 2, 5, 10, 17,26......## this is how i did it: 2, 5, 10, 17, 26 can form a difference of 3,5, 7,9... whose nth term is 1 + 2n. now how can apply this to the original sequence to get the nth term?

Homework Statement I was wondering how I would go about showing that (an) is monotone decreasing given that an = 1/√n. I believe I have to show an ≥ an+1, but I'm not sure how to go about doing that.

Homework Statement A program finds the area under the Gaussian Distribution Curve between ±σ using Simpsons Rule. Modify the program to investigate the effect of the number of strips. Do this by using a DO loop in the main program for the following sequence of number of strips (n); n-2, n-4...

This sequence is similar to Sloane's #A010551; one further multiplication is made... n= 0 1 2 3 4 5 6 7 8 9 10 11 12 1 1 1 1 2 4 8 24 72 216 864 3456 13824... Find a formula for the nth term.

Hello! (Smile) We have a sequence of numbers $A=(a_1, a_2, \dots, a_n), n \geq 3$, for which, it stands the following: $$\forall i \text{ with } 2 \leq i \leq n-1:$$ either $$ 1. a_{i-1}<a_i \text{ and } a_i>a_{i+1}$$ or $$2. a_{i-1}>a_i \text{ and } a_i<a_{i+1}$$ I have to write an...

Homework Statement Show if this sequence (with n=1 to infinity) diverge or converge Homework Equations [/B] The Attempt at a Solution If I use the Limit Comparison Test: compare with so you get that equals lim n -> inf => inf. Can I use the Test like this? What does this...

Hello! (Smirk) Consider that a sequence of 10 push and 10 pop operations is executed at an initially empty stack. The ten push operations place the numbers $0,1, \dots, 9 $, in an ascending order, in the stack. It is possible, that between the push operations we can have some pop operations...

Hey! (Wave) Sentence: The p-adic numbers are complete with respect to the p-norm, ie every Cauchy sequence converges. Proof: Let $(x_i)_{i \in \mathbb{N}}$ a Cauchy-sequence in $\mathbb{Q}_p$. We want to show that, without loss of generality, we can suppose that $x_i \in \mathbb{Z}_p$.Let...

given :$(a_{n+1}+3)(2-a_n)=6$ $(a_n\neq 0, a_1=1)$ please find :$\sum_{n=1}^{100} \dfrac {1}{a_n}$

Need help on these questions I haven't been introduced yet. Please look at image I inserted in.

Is this possible? It seems like it should be but, it's difficult to find an explicit relationship between a general function of one variable x (let's say we are only interested in functions that decay to zero as they go to plus or minus infinity) it seems like summing a bunch of gaussians of...

prove that limit of a cuachy sequence of integers is an integer

Hi! I'm new in the forum and I bring a challenge Here is the problem: For a positive integer $$x$$, denote its n-th decimal digit as $$d_{n}(x)$$, with $$d_{n}(x)\in \{0,1,\ldots ,9\}$$ so $$x=\displaystyle\sum_{i=1}^{+\infty}d_{i}(x)10^{i-1}$$. Let $$(a_{n})_{n\in \Bbb{N}}$$ be a squence...

Homework Statement Find the next three numbers in this sequence... 4, 16, 21, 21, 18, 14, 11, ... 2. The attempt at a solution The difference between each term is 12, 5, 6, 0, -3, -4, -3 but I can't see a pattern and I am completely stuck... Any help pls?

I'm not sure if this is true or not. but from what I can gather, If the set of Natural numbers (divergent sequence) {1, 2, 3, 4, 5,...} is broken up to say {1}, is this a subsequence that converges and therefore this statement is true?

I've been very confused with this proof, because if a sequence { 1, 1, 1, 1, ...} is convergent and bounded by 1, would this be considered to be a Cauchy sequence? I'm wondering if this has an accumulation point as well, by using the Bolzanno-Weirstrauss theorem. I really appreciate the help...

Hello, I am well aware of the ratio method, and the sum = 1/(1-r) but I want to try this method. I am trying to understand this: \displaystyle \sum_{n=1}^{\infty} e^{-n} using integrals, what I have though: = \displaystyle \lim_{m\to\infty} \sum_{n=1}^{m} e^{-n} = \displaystyle...