Sequence Definition and 1000 Threads

In the attached picture, how is it justified to factor out an 'i' in the exponent of the 2 and the 5?

Homework Statement Consider the sequence given by b_{n} = n - \sqrt{n^{2} + 2n}. Taking (1/n) \rightarrow 0 as given, and using both the Algebraic Limit Theorem and the result in Exercise 2.3.1 (That if (x_n) \rightarrow 0 show that (\sqrt{x_n}) \rightarrow 0), show \lim b_{n} exists and find...

Given that $S=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4} + ...+\dfrac{1}{99}-\dfrac{1}{100}$. Prove that $\dfrac{1}{2}<S<1$.

I figured it out... how do I remove this question?

Homework Statement The sum of first three numbers of the arithmetic sequence is 54. If you subtract 3 from the first one, leave the second one unchanged and add 12 to the third one you get the first three numbers of the geometric sequence of the form ##ar + ar^2 + ar^3 + ... ar^n ## Find r...

A sequence as follows : $0,1,-1,2,3,-2,4,5,6,-3,7,8,9,10,-4,11,12,13,14,15,16,-5,--------$ please find :$a_{2008}=?$

Hi, I am trying to prove that every convergent sequence is Cauchy - just wanted to see if my reasoning is valid and that the proof is correct. Thanks! 1. Homework Statement Prove that every convergent sequence is Cauchy Homework Equations / Theorems[/B] Theorem 1: Every convergent set is...

Let $\left\{{x}_{n}\right\}$ be a sequence...İf $\left\{{x}_{2n}\right\}$ is caucy sequence, can we say that $\left\{{x}_{n}\right\}$ is cauchy sequence ?

Hi guys, I attempted to prove this theorem, but just wanted to see if it a valid proof. Thanks! 1. Homework Statement Prove that x is an accumulation point of a set S iff there exists a sequence ( s n ) of points in S \ {x} that converges to x Homework Equations N * ( x; ε ) is the x -...

Denote with ${\varPsi}_{st}$ the family of strictly nondecreasing functions ${\Psi}_{st}:[0,\infty)\to [0,\infty)$ continuous in $t=0$ such that ${\Psi}_{st}=0$ if and only if $t=0$ ${\Psi}_{st}(t+s)\le {\Psi}_{st}(t)+{\Psi}_{st}(s)$. Definition: Let $\left(X,d\right)$ be a metric space and...

If a sequence of operators \{T_n\} converges in the norm operator topology then: $$\forall \epsilon>0$$ $$\exists N_1 : \forall n>N_1$$ $$\implies \parallel T - T_n \parallel \le \epsilon$$ If the sequence converges in the strong operator topology then: $$\forall \psi \in H$$...

Homework Statement I'm trying to find out whether or not this sequence diverges or converges. If it converges, then what's the limit. {4+sin(1/2*pi*n)} The Attempt at a Solution This one is a bit confusing to me since sin oscillates between 1 and -1. So if you plug in (pi*infinity)/2, that...

Greetings all, I am registering for spring 2016 courses and have one question. I can pick up a math course and I have the option between two courses: 430 Formal Logic vs. 481 Applied Partial Differential Equations. I am a math and physics double major. Course list and description...

If I open a new deck of cards the sequence is known. If I shuffle them, the sequence is unknown. If I then memorize the sequence, it is again known. State 1: New deck State 2: Shuffled deck State 3: Shuffled deck after I have memorized the sequence State 4: Re-shuffled deck I have computed...

Homework Statement Given a sequence (x_n), x_n > 0 for every n\in\mathbb{N} and \lim\limits_{n\to\infty} x_n = L > 0, show that \ln x_n\to \ln L when n\to\infty. Homework EquationsThe Attempt at a Solution As logarithm function is an elementary function, meaning it is continuous in its domain...

Hey Guys, So I'm currently factoring polynomials. As you all know,when trying to factor a polynomial with 4 terms,the answer can be the difference of a trinomial and monomial squares. So I had the problem 4a^2-4ab+b^2-c^2 I thought(book doesn't explain it well) that your first step was to...

Homework Statement Show, from the definition of what it means for a function to converge to a limit, that the sequence ##\left\{x^t\right\}_{t=1}^{\infty}## with ##x^t = \frac{2t+5}{t^2+7}## converges to ##0## as ##t## goes to infinity. Homework Equations A sequence converges to ##x^0 \in X##...

Homework Statement Suppose a coin is tossed 14 times and there are 3 heads and 11 tails. How many such sequences are there in which there are at least 6 tails in a row? The Attempt at a Solution I will treat the sequence of coin tosses as a "word" where each letter is a toss and is either an...

The explanation of a continuous Markov process X(t) defines an indexed collection of sigma algebras by \mathcal{F}_t = \sigma\{ X(s): s < t\} and this collection is said to be increasing with respect to the index t . I'm trying to understand why the notation used for set inclusion is...

Is it possible to come up for a sequence property or sequence sum property for ΣnK=1K^-1 If so, what other sequence properties that are not commonly seen are there?

Hi! If I have a sequence that its first 4 terms are: 30, -31, +32, -32 The pattern is geometric sequence but has alternating signs.. How can I find its sum .. I know it is composed of 2 sequences .. However, when I try to separate the 2 sequences .. I get them of different "lengths" In...

My Questions: 1) İn both sides of inequality of (*) why we use "n", that is, why we do multiplication with "n" ? 2) in (**) by Letting $n\to\infty$ we obtain $\lim_{{n}\to{\infty}} n\left[d\left({T}^{n}x,{T}^{n+1}x\right)\right]{}^{r}=0$ How this...

Homework Statement : [/B]Prove that if xn is a sequence such that |xn - xn+1| ≤ (1/3n), for all n = 1,2,..., then it converges.Homework Equations : [/B]The definition of convergence.The Attempt at a Solution :[/B] I attempted to prove this by induction, so I am clearly far off the mark here...

Homework Statement : [/B]Let a = sup S. Show that there is a sequence x1, x2, ... ∈ S such that xn converges to a.Homework Equations : [/B]I know the definition of a supremum and convergence but how do I utilize these together?The Attempt at a Solution :[/B] Given a = sup S. We know that a =...

The pth term of an airthmetic progression is 1/q and qth term is 1/p. What is the sum of pqth term?

Let $a_n$ be the $n$th term of the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, ...,$ constructed by including the integer $k$ exactly $k$ times. Show that $a_n=\lfloor\sqrt{2n}+\frac{1}{2}\rfloor$. (Hints only as this is an assignment problem.)

Hello Physics Forums community! I've been struggling for a while with this one ; so basically a sequence a_n is given to us such that the sequence b_n defined by b_n = pa_n + qa_(n+1) is convergent where abs(p)<q. I need to prove a_n is convergent also. Any hint would be of so much help, thank...

Homework Statement Prove that if a bound sequence ##\left\{ { X }_{ a } \right\} ## is divergent then there are two sub sequences that converge to different limits. Homework Equations None. The Attempt at a Solution Ok so I am not sure if my attempt for a solution is correct or not, but I...

Let $X={\ell}^{\infty}:=\left\{u\in{\ell}^{2}\left(R\right):\left| {u}_{k} \right|\le\frac{1}{k}\right\}$ and $T:{\ell}^{\infty}\to{\ell}^{\infty}$, defined by $T{u}_{k}=\frac{k}{k+1}{u}_{k}.$. Then 1) ${\ell}^{\infty}$ is a compact metric space, 2) $T$ is not a contraction...

Homework Statement Show that the sequence {xn}: xn := (21/1 - 1)2 + (21/2 - 1)2 + ... + (21/n - 1)2 is convergent. Homework EquationsThe Attempt at a Solution If n > m, |xn - xm| = (21/n - 1)2 + (21/(n-1) - 1)2 + ... + (21/(m+1) - 1)2 < (21/n)2 + (21/(n-1))2 + ... + (21/(m+1))2 < (21/(m+1))2 +...

Homework Statement prove: \lim_{n\rightarrow \infty} {\frac{n!}{2^n}}=\infty Homework Equations Def. of a limit The Attempt at a Solution I would like to know if my solution is right or not. I think it is right but I would like to get a feedback. Please do not give me the answer, just...

I see that derivative of y with respect to x is just like the ratio of y over x. But, Why Un (the formula to find nth term) is not the derivative of Sn (the sum of sequence formula) ?? For example, 1 2 5 10 -> y = x2+1 +1...

İn some articles, I see something... For example, Let we define a sequence by ${x}_{n}=f{x}_{n}={f}^{n}{x}_{0}$$\left\{{x}_{n}\right\}$. To show that $\left\{{x}_{n}\right\}$ is Cauchy sequence, we suppose that $\left\{{x}_{n}\right\}$ is not a Cauchy sequence...For this reason, there exists a...

Let $\left\{{a}_{n}\right\}$ be a nonnegative, non-increasing sequence and convergence to $a \ge 0$. Can we say that ${a}_{n}\ge a$ for all n $\in \Bbb{N}$ ? Also, if $\left\{{a}_{n}\right\}$ is a nonnegative, decreasing sequence and convergence to $a \ge 0$. Can we say that ${a}_{n}> a$ for...

Given a sequence, how to check if it converges? Assume the sequence is monotonic but the formula that created the sequence is unknown. My first thought was if: seq(n+2) - seq(n+1) < seq(n+1) - seq(n) , is always true as n->infinity then it is convergent. Or in other words, if the difference...

Let $\left(E,d\right)$ be a complete metric space, and $T,S:E\to E$ two mappings such that for all $x,y\in E$, $d\left(Tx,Sy\right)\le M\left(x,y\right)-\varphi\left(M\left(x,y\right)\right)$, where $\varphi:[0,\infty)\to [0,\infty)$ is a lower semicontinuous function with...

Homework Statement 1. Let ##x_n = \frac{n^2 - n}{n} ## does ##x_n## converge or diverge? 2. Let ##x_n = \frac{(-1)^n +1}{n} ## does ##x_n## converge or diverge Homework Equations A sequence converges if ##\forall \epsilon > 0##, ##\exists N \in \mathbb{N} ## such that ##n\geq N ## implies ##...

$\newcommand{\Z}{\mathbb Z}$. Question: Let $m$ and $n$ be positive integers. What are all the abelian groups $A$ such that there is a short exact sequence $0\to \mathbb Z/p^m\mathbb Z\to A\to \mathbb Z/p^n\mathbb Z\to 0$. It is clear that any such abelian group $A$ has cardinality $p^{m+n}$...

to prove that $\left\{{x}_{n}\right\}$ is Cauchy seqeunce we use a method. I have some troubles related to this method. Please help me... $\left\{{c}_{n}\right\}$=sup$\left\{d\left({x}_{j},{x}_{k}\right):j,k>n\right\}$.Then $\left\{{c}_{n}\right\}$ is decreasing. If ${c}_{n}$ goes to 0 as n...

I have been writing c# for fun a few weeks but i always run into problem with for circles I wrote a simple code to show me first n numbers of fibonacci sequence but i don't know where the problem is I had similar issues with it before ... Even if the code is wrong then at least it should give...

$y_0=k$ where $k$ is a constant. $x_{n+1}=30-\dfrac{y_n}{2}$ $y_{n+1}=30-\dfrac{x_{n+1}}{2}$ Prove that $(x_n, y_n)$ converges to $(20, 20)$ for all values of $k$. My attempt: I wrote a computer program and verified this for a few values of $k$. But I don't know how to prove that $x_n$ and...

Homework Statement { 7/6 , 45/54 , 275/648, 1625/9720 ... } Problem from a practice exam. My instructions are to find an explicit formula for the above sequence.. The Attempt at a Solution Let An =7/6 , 45/54 , 275/648, 1625/9720 ... Allow An = Bn / Cn Bn = 7, 45, 275, 1625... (2n+5)...

I literally don't know how to solve this one. I hope you can help me. :)

Hello let be a finish normed vectoriel space $$(E, ||.||)$$, and $$u \in L(E) / ||u|| \leq 1$$. 1) Show that $$Ker(u - Id) = Ker((u - Id)^{2})$$. For $$\subset$$ it's obvious, but for $$\supset$$ I don't know. I suppose $$\exists x_{1} \in Ker((u - Id)^{2})\Ker(u - Id)$$. So I can say that...

Homework Statement Hi, I am reviewing a practice exam for my course and I am a bit stuck. "Assume that a sequence of partial sums (s_n) converges, can we also then say the sequence a_n is convergent? Does this statement go both ways? Answer: Yes, yes" The Attempt at a Solution On our exam...

Mod note: Moved from the homework section. 1. Homework Statement If σ > 0 and base b > 1 then prove that ##log n/n^σ## is a null sequence. This is not really a homework since i am self studying Konrad Knopp book about infinite series and i wanted to see different ideas and perspectives on...

Hello, today I have a question. If uni.cont. function sequence fn on (0,1) is uni.conv. to f on (0,1), then f is uni.cont. on (0,1). The above is true. The wonder I have is...May I prove the above by this way? This way: fn is uniform continuous on (0,1). So Fn is continuous on [0,1] (by...

Let $\{p_n\}$ be a nonnegative nonincreasing sequence and converges to $p \ge 0$. Let $f : [0,\infty)\to[0,\infty)$ be a nondecreasing function. So, since f is a nondecreasing function, $f(p_n)>f(p)>0$. How did this happen?

Without buying a new camera, can anyone suggest a way to set up a time lapse of an artist working on a piece of art, like a sculpture? You know, put the camera on a stand pointing at the artwork, and have it take one pic, say, each second? Either a point & shoot camera or a DSLR.

Homework Statement Hi, I've been solving Calculus Deconstructed by Nitecki and I've been confused by a particular lemma in the book. Namely: If a sequence is eventually bounded, then it is bounded: that is, to show that a sequence is bounded, we need only find a number γ ∈ R such that the...