Sequence Definition and 1000 Threads

Homework Statement I was given this homework problem: Show that if ##a_1,a_2, ... ,## is a sequence of real numbers that converges to ##a##, then lim_{n\to \infty}\frac{\sum^n_{k=1} a_k}{n}=a. I was provided a solution but my book never went over such examples or the concrete steps to solve...

Hello! Please help me to solve following exercise (2.5.8) from Elementary Real Analysis by Thomson-Bruckner: Suppose that a sequence \{s_n\} of positive numbers satisfies the condition s_{n+1} > \alpha s_n for all ##n## where ##\alpha>1.## Show that ##s_n \to \infty.## I can't prove...

Homework Statement Define a1=1, and for every n>1, an+1 = an + \frac{1}{an}. Prove that 20 < a200 < 24The Attempt at a Solution I tried a few things to no avail. First, I showed that this is an increasing function by showing an+1 > an. I tried finding a limit, by saying if...

an(n in subindex)=(1/2)*n^2-3n+5/2, when n ≥1 Is number 10 member of that sequence? what about number 6?Create equation to solve it. If someone can help with this problem please, it will be much appreciated!

1. the nsider, for n → 1, the sequence an given by an = n log (n/n+1) Determine the limit of the sequence as n→∞, If it exists , or explain why the sequence diverges. In your answers include the names of any rules, theorems or limits you have used. 2. Homework Equations 3. The Attempt at a...

I have no idea how to solve this equation, its in my homework... i know the formula to find the nth term(tn=ar^n-1) but don't know how to solve this: The difference between the first term and second term in a geometric sequence is 6.The difference between the second term and the third term is...

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

the main question here is that can a sequence * arithmetic * be correct if the difference is also changing in terms of a geometric sequence ?\ now look at this sequence 0.33,0.3333,0.333333 now if we calculate the difference between the first two terms its 0.0033 between the second and...

Is it possible, using the Collatz hailstone sequence to ever start at a number n and end up with 2n at some point later in the sequence for values greater than 2? Can you have a sequence that goes n . . . 2n (I don't care what any of the exact values are, I want to know if using variables it is...

Consider the sequence of positive integers which satisfies a_n=a_{n-1}^2+a_{n-2}^2+a_{n-3}^2 for all $n \ge 3$. Prove that if $a_k=1997$, then $k \le 3$.

Homework Statement A bored student enters the number 0.5 in her calculator, then repeatedly computes the square of the number in the display. Taking A0 = 0.5, find a formula for the general term of the sequence {An} of the numbers that appear in the display, and find the limit of the sequence...

Homework Statement Find the limit of n22n/(n!) Homework Equations The Attempt at a Solution First I expand out 2n/(n!) = (2/1)(2/2)(2/3)(2/4)...(2/n) which gets increasingly small as n increases. Now, where does the n2 fit into this? I know the limit to be 0 but I can't get...

Homework Statement Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges find its limit. an = (1*3*5*...*(2n-1))/(2n)n Homework Equations lim n->infinity an = L The Attempt at a Solution The answer in the book shows: 1/2n *...

A sequence of positive integers is defined as follows: The first term is 1. Then take the next two even numbers 2, 4. Then take the next three odd numbers 5, 7, 9. Then take the next four even numbers 10, 12, 14, 16 and so on. Find the nth term of the sequence.

Homework Statement Let a be a real number such that 1 < a < 2 ,{an} is a sequence defined by a1=a, an+1=|an|-1 (n=1,2,3...) And put sn=a1+a2+a3+...an i)Find the a4,a5,a6,a7 i)Find a4,a5,a6,a7 The Attempt at a Solution I don't even know how to start.Maybe someone could give me an idea.

I should delete one unthanked post to get , 444 666 999 that was just close (Rofl)

Find \sum_{x=0}^{101}\frac{\frac{2x}{101}-1}{\frac{3x^2}{10201}-\frac{3x}{101}+1}.

Homework Statement find the limit n\rightarrow∞ of 10n/ n! Homework Equations L hospital rule The Attempt at a Solution took log and separated the num and denom as: n ln10-ln(n!) n ln10-n ln(n)+n 1/n ( ln10 - ln(n)+1) now i...

Consider the following sequence of statements: $$ S_1: \text{at least 1 of the statements }S_1-S_n \text{ is false}\\ S_2: \text{at least 2 of the statements }S_1-S_n \text{ are false}\\ \vdots \\ S_n: \text{at least } n \text{ of the statements }S_1-S_n \text{ are false} $$ Where $n$ is some...

Homework Statement Let (M,d) be a complete metric space and define a sequence of non empty sets F1\supseteqF2\supseteqF3\supseteq such that diam(Fn)->0, where diam(Fn)=sup(d(x,y),x,y\inFn). Show that there \bigcapn=1∞Fn is nonempty (contains one element). Homework Equations The...

Hello everyone, I just wanted to make a post thanking all of those who have helped me over the past couple of years with my physics and math questions, since it would be hard to reach every individual. I am now finally done with the lower division physics sequence and managed to get an A in...

Homework Statement Is the sequence \frac{1}{1}, \frac{1}{2}, \frac{1}{3} , \frac{1}{4}...\frac{1}{n} arithmetic or geometric? Homework Equations Common difference and Common ratio formulas The Attempt at a Solution I found the common difference from a_{2} - a_{1} =d_{1} and common...

Homework Statement Let \left\{E_{k}\right\}_{k\in N} be a sequence of measurable subsets of [0,1] satisfying m\left(E_{k}\right)=1. Then m\left(\bigcap^{\infty}_{k=1}E_{k}\right)=1. Homework Equations m denotes the Lebesgue measure. "Measurable" is short for Lebesgue-measurable. The Attempt...

Homework Statement prove the sequence ## \frac{n!^{1/n}}{n} ## converges and find its limit Homework Equations n/a The Attempt at a Solution Ok since this question was in a section having to do with the ratio test, I am making an educated guess that we are suppose to show...

Here is the question: Here is a link to the question: Find the limit of sequence? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

My sequence is a function f:Z+→{H,T} to represent heads and tails. It is defined by f(1),f(2),f(3)f(4),f(5)f(6), ... = H,T,HH,HT,TH,TT,HHH,THH,HTH,..., where you can see that I'm just taking all heads-tails sequences of length 1, then of length 2, etcetera. This ensures (1) that the...

{Edit: as of 3:55 eastern time, made corrections to tex and itex mistakes}Is this all kosher in terms of demonstrating accuracy and comprehension of the notation {a_{1} + a_{2}...} = \lim_{n\rightarrow ∞ } \sum_{n=1}^{n} a_{n} So the lower case represents sequences and upper case represents...

How to find nth term of the sequence 6+13+24+39+...

Hi all, I've only just started MIPS and have been stuck on this introductory lab for a while now. I have 8 LEDs each lit by there corresponding bit being a 1. Hence 0010 0101 at the input would light up LEDS 1, 3 and 6. I need to make LEDs 1-5 light up one at a time and extinguish the...

$f(5)=10$ $f(n+3)=\dfrac {f(n)-1}{f(n)+1},\,\, for \,\, all \,\,n\in N$ find f(2009)

I want to prove the sequence a(n) = n diverges, directly, without the aid of any theorems. Naturally, I try to prove this by contradiction. Here's my attempt: Let L be a real number such that a(n) converges to L. Then for all e>0, there exists a natural number N s.t. any n>N implies d(a(n) -...

Homework Statement Well, this thread is purposely to clarify my question on the use of L'Hospital's rule for sequence. As I have read from the calculus book, the sequence can be defined such that f:N → ℝ with function f(n)= an where n inside natural numbers,N. So, we cannot apply L'...

Homework Statement Show that the sequence (xn)n\inN \inZ given by xn = Ʃ from k=0 to n (7n) for all n \in N is a cauchy sequence for the 7 adic metric. Homework Equations In a metric space (X,dx) a sequence (xn)n\inN in X is a cauchy sequence if for all ε> 0 there exists some M\inN such...

Homework Statement Let g(x)= (2/3)*(x+1/(x^2)) and consider the sequence defined by pn= g(pn-1) where n≥1 a) Determine the values of p0 \in [1,2] for which the sequence {pn} from 0 to infinity converges. b) For the cases where {pn} converges (if any), what is the rate of convergence...

Suppose (an) is sequence in the metric space X and define Tn={ak:k>n} and diamT=sup{d(a,b):a,b elements of T}. Prove that (an) is Cauchy if and only if diam Tn converges to zero. In what metric spacee does Tn converge? I assumed in (ℝ,de) but this is confusing since the diam of T is...

Homework Statement Use sandwich Rule to find the limit lim n> infinity (a_n) of the sequences, for which the nth term, a_n, is given. Homework Equations ^{lim}_{n\rightarrow∞}\frac{n!}{n^{n}} The Attempt at a Solution I know by just looking at it, n^n Approaches infinity much...

Calculus 2: Sequence Limits Question to the power n?? Homework Statement Find the limits (if it exists) to decide which sequences, whose nth term is given below. Homework Equations (\frac{3^{n}-4^{n}}{3n^{2}+4^{n}+7}) The Attempt at a Solution I've done a few of these but as Soon as the...

Homework Statement Hi I've come across the term lim inf ##f_n## in my text but am not sure what it means. ##\lim \inf f_n = \sup _n \inf _{k \geq n} f_k## In fact, I am not sure what is supposed to be the output of lim inf f? That is, is it supposed to return a real-valued number, or a...

Homework Statement The question : http://gyazo.com/7eb4b86c61150e4af092b9f8afeaf169 Homework Equations Sup/Inf axioms Methods of constructing sequences ##ε-N## ##lim(a_n) ≤ sup_n a_n## from question 5 right before it. I'll split the question into two parts. The Attempt at a...

Homework Statement Suppose that E is contained in ##\mathbb{R}## is a nonempty bounded set and that ##\sup E## is not in E. Prove that there exists a strictly increasing sequence ##\left\{x_n\right\}## that converges to ##\sup E## such that ##x_n \in E## for all n in ##\mathbb{N}##...

Homework Statement http://gyazo.com/d59c730eb9b18dda4504a5fe118c7213 Homework Equations Limit and supremum. The Attempt at a Solution (a) Let : ##b_n = a_n - b## so that ##b_n ≤ 0## Now, ##lim(b_n) = lim(a_n - b) ≤ 0 \Rightarrow a - b ≤ 0 \Rightarrow a ≤ b## Q.E.D (b)...

I'm having problems generalizing a number sequence. It involves a couple parameters that I will attempt to explain and show by example. Basically I have a program that takes 3x3 raster segments of a much bigger matrix. The matrix is a logical matrix. What I do is take the a 3x3 raster segment...

1. Determine whether the sequence with the given nth term is monotonic & bounded. a_n = (n) / (2^(n+2))2. b_n < b_n+1 3. (n) / (2^(n+2)) < (n+1) / (2^(n+3)) I multiply both side by (2^(n+2)) and (2^(n+3)) (n)(2^(n+3)) < (n+1)(2^(n+2)) Then i distribute and got: (n)(2^(n+3)) < (n)(2^(n+2))...

Homework Statement Prove that $$\frac{x_n^2 - e}{x_n} \rightarrow 1-e$$ as ##n \rightarrow \infty##, provided ##x_n \rightarrow 1## as n ##\rightarrow \infty##. The Attempt at a Solution The above holds if ##\,\forall\, \epsilon > 0 \,\exists \, N\, \in\, \mathbb{N}## such that if n...

Could you help me to find the general term of the sequence: ## 1 , \frac{5}{3} , 1 , \frac{15}{17} , 1 , \frac{37}{35} , 1 , \frac{63}{65} ,... ## Thank you!

Homework Statement Two insulating materials of thermal conductivity K and 2K respectively are to be used as two layers of insulation for lagging a pipe carrying hot fluid. If the radial thickness both the layers is to be same, then - i) the first material (thermal conductivity K) should be...

Homework Statement I have a problem where I need to know if I can make a sequence for this data to the nth term. The first term, however, isn't common. Is there any way to somehow make it a sequence? At dose 1 y(1)=De^(0*12*k)/(1-e^(0*12*k)) at dose 2 y(2)=D*e^(1*12*k)/(1-e^(12k)) +...

write the terms a1,a2,a3,a4 of the following sequence. an+1=0.4an+330, a0=550 everytime I get 550 for a1 a2 a3 and a4 is that correct or am I doing it wrong.

Calc 2 will require some basic knowledge of sequences and series. Since this topic has never been covered in any of my past math classes, I am currently learning about sequences and series from scratch. On youtube, I found a video that contains an explanation of the General Term of the...

Homework Statement The sum S n of the first n terms of a geometric sequence whose nth term is u n is given by 7n-an / 7n Where a > 0 Find an expression for un Find the first term and the common ratio of the sequence Consider the sum to infinity of the sequence Determine...