Sequence Definition and 1000 Threads

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 -...

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...

Hi guys, I am trying since a while to put in equation the evolution of a star's central density, temperature as it leaves the main sequence but has not reached yet the burning of Helium. So there is no nuclear reaction in the centre and the core is slowly collapsing. Does anyone have some...

The sequence that I'm working with is sort of monotonic. It's monotonic most of the time, sometimes there's one number that ruins the trend like 0.0001001 in 0.1, 0.01, 0.001, 0.0001001, 0.0001, ... but those are very rare. Btw, the sequence above is just an example. Is Richardson...

I have been given the following sequence: an+1 = an + (1/2)n+1, n>=0 with a0 = 1. I am trying to express a5 in terms of 1/2(a4) . I have started by writing out the first few terms however i am still struggling with getting a5. Can anyone please help me out with this? Thanks in advance!