Sequences Definition and 576 Threads

Homework Statement Determine the triangles where the sides are consecutive elements of a geometric sequence and the angles are consecutive elements of an arithmetic sequence. Homework Equations The Attempt at a Solution I don't really know how to approach this problem, what the solution would...

I have a Dover edition of Louis Brand's Advanced Calculus: An Introduction to Classical Analysis. I really like this book, but find his proof of limit laws for sequences questionable. He first proves the sum of null sequences is null and that the product of a bounded sequence with a null...

Homework Statement Let ##K\neq\emptyset## be a compact set in ##\Bbb{R}## and let ##c\in\Bbb{R}##. Then ##\exists a\in K## such that ##\vert c-a\vert=\inf\{\vert c-x\vert : x\in K\}##. 2. Relevant results Any set ##K## is compact in ##\Bbb{R}## if and only if every sequence in ##K## has a...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Corollary 4.2.6 ... ... Corollary 4.2.6 reads as follows: Bland gives a statement of Corollary 4.2.6 but does...

Hi Physics Forums, I have a problem that I am unable to resolve. The sequence ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}## of positive integer powers of ##\mathrm{sinc}(x)## converges pointwise to the indicator function ##\mathbf{1}_{\{0\}}(x)##. This is trivial to prove, but I am struggling to...

I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 3.2 Exact Sequences in [FONT=MathJax_Main]Mod[FONT=MathJax_Math]R ... ... I need some help in order to fully understand the proof of Proposition 3.2.7 ... Proposition 3.2.7 and its proof read...

I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 3.2 Exact Sequences in ##\text{Mod}_R## ... ... I need some help in order to fully understand the proof of Proposition 3.2.7 ... Proposition 3.2.7 and its proof read as follows: In the above...

I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 3.2 Exact Sequences in $$\text{Mod}_R$$ ... ... I need some further help in order to fully understand the proof of Proposition 3.2.6 ... Proposition 3.2.6 and its proof read as follows: In the...

I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 3.2 Exact Sequences in $$\text{Mod}_R$$ ... ... I need some help in order to fully understand the proof of Proposition 3.2.6 ... Proposition 3.2.6 and its proof read as follows: In the above...

I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 3.2 Exact Sequences in ##\text{Mod}_R##... ... I need some help in order to fully understand the proof of Proposition 3.2.6 ... Proposition 3.2.6 and its proof read as follows: In the above...

Homework Statement Let ##x\in\Bbb{R}## such that ##x\neq 0##. Then ##x=LIM_{n\rightarrow\infty}a_n## for some Cauchy sequence ##(a_n)_{n=1}^{\infty}## which is bounded away from zero. 2. Relevant definitions and propositions: 3. The attempt at a proof: Proof:(by construction) Let...

Can someone help me with these. These are the last 5 questions that I have to do and if I get them right I pass maths.

Could someone check if my answers are right and help me with question 5?

Homework Statement http://sites.math.rutgers.edu/~ds965/temp.pdf (NUMBER 2)[/B]Homework Equations I do not understand the alternating part for the second problem and the recursive part for the first problem.The Attempt at a Solution The first answer I got was first by writing out the...

If I have monotonic sequence, would it suffice to analyze |a(n)-a(n-1)| as n gets large? I know for Cauchy sequences, you have to analyze every term after N, but for monotonic sequences that are also Cauchy, can you just analyze the difference between consecutive terms?

Homework Statement "Let ##E \subset ℝ##. Prove that ##E## is closed if for each ##x_0##, there exists a sequence of ##x_n \in E## that converges to ##x_0##, it is true that ##x_0\in E##. In other words, prove that ##E## is closed if it contains every limit of sequences for each of its...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Theorem 1.6.5 (Completeness of $$\mathbb{R}^n$$) ... Duistermaat and Kolk"s Theorem 1.6.5 and...

Homework Statement [/B] The proposition that I intend to prove is the following. (From Terence Tao "Analysis I" 3rd ed., Proposition 6.1.7, p. 128). ##Proposition##. Let ##(a_n)^\infty_{n=m}## be a real sequence starting at some integer index m, and let ##l\neq l'## be two distinct real...

Homework Statement For some background, this is an advanced calculus 1 course. This was an assignment from a quiz back early in the semester. Any hints or a similar problem to guide me through this is greatly appreciated! Here is the problem: Find a convergent subsequence of the sequence...

I will first summarize the construction of ordinal numbers and introduce the definition of the binary Veblen function and of the notion of fundamental sequence. Ordinal numbers start with natural numbers 0, 1, 2, 3, ... which are followed by ## \omega ## which represents the "simple" infinity...

Dear ALL, My last Question of the Day? Let b1 and b2 be a sequence of numbers defined by: $$b_{n}=b_{n-1}+2b_{n-2}$$ where $b_1=1,\,b_2=5$ and $n\ge3$ a) Write out the 1st 10 terms. b) Using strong Induction, show that: $$b_n=2^n+(-1)^n$$ [FONT=garamond]Many Thanks John C.

I am familiar with the following formulation of the principle of recursive definition. Now, in certain proofs in analysis, there are times where a recursive definition for a function is used. Here are two examples. ##\textbf{Proof:}## Let ##p## be a limit point of ##E##, let ##\epsilon > 0##...

I am reading "Introduction to Real Analysis" (Fourth Edition) b Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 2: Sequences and Series ... I need help in fully understanding Example 3.4.3 (b) ...Example 3.4.3 (b) ... reads as...

I am reading "Introduction to Real Analysis" (Fourth Edition) b Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 2: Sequences and Series ... I need help in fully understanding Example 3.4.3 (b) ... Example 3.4.3 (b) ... reads as follows: In the above text from Bartle and...

In his book: Introduction to Real Analysis, Manfred Stoll does not prove parts (a) and (b) of Theorem 2.2.6 on the limits to certain special sequences ... I am having trouble getting started on the proof of part (a) ... can someone please help me to make a meaningful start to the proof ...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with a part of Exercise 2.2.7 Part (1) ... ... Exercise 2.2.7 Part (1) reads as follows:I have managed a solution to this...

Homework Statement I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with a part of Exercise 2.2.7 Part (1) ... ... Exercise 2.2.7 Part (1) reads as follows: I have managed a...

Hi guys, I am not sure if my understanding of subsequence is right. For example I have sequence {x} from n=1 to infinity. Subsequence is when A) I chose for example every third term of that sequence so from sequence 1,2,3,4,5,6,7,... I choose subsequence 1,4,7...? B) Or subsequence is when I...

Hey! :o I want to check which of the following sequences converges and from those that don't converge I want to check if it has a convergent subsequence. $\displaystyle{1, 1-\frac{1}{2}, 1, 1-\frac{1}{4}, 1, 1-\frac{1}{6}, \ldots}$ $\displaystyle{1, \frac{1}{2}, 1, \frac{1}{4}, 1...

So the definition of a bounded sequence is this: A sequence ##(x_{n})## of real numbers is bounded if there exists a real number ##M>0## such that ##|x_{n}|\le M## for each ##n## My question is pretty simple. How does one choose the M, based on the sequence in order to arrive at the...

Homework Statement Give the example and show your understanding: [1][/B].Lets define some property of a point of the function: 1. Point is a stationary point 2. Point is a max/min of a function 3. Point is a turning point of a function If possible name a function whose point has properties of...

So we just recently did accumulation points in my maths class for chemists. I understood everything that was taught but ever since I was trying to find a reasonable explanation if the sequence an = (-1)n has 2 accumulation points (-1,1) or if it doesn't have any at all. I mean it's clear that...

The pattern above will continue for all values of the harmonic sequence. Will a destination point be reached for any value of θ where 0 ≤ θ < 2𝜋? (I know it won’t for θ = 0) Is there a function which contains the set of all destination points?

I've started Book of Proof, the first chapter of which is an intro to sets. Q.1 Is there any particular way to approach these kinds of problems, other than using intuition / trial & error? I tend to have some difficulty in working out the best way to express the general term of a sequence, for...

Say that we have a sequence defined by the mth degree polynomial, ##a_n=\displaystyle \sum_{k=0}^{m}c_kn^k##. I found the following formula which is a recursive representation of the same sequence: ##\displaystyle a_n =\sum_{k=1}^{m+1}\binom{m+1}{k} (-1)^{k-1}a_{n-k}##. I'm curious as to why...

Part 1 of this little problem is here: http://mathhelpboards.com/linear-abstract-algebra-14/little-problem-exact-sequences-19368.html. This is part 2: who can formulate and prove the dual statement?

I have this little problem on exact sequences, I want to check with you. We have two R-maps $f:A\to B$ and $g:B\to C$ of left R-modules. And we have an isomorphism $B/\mbox{im} f\cong C$ "induced by g" . Then prove that the sequence $S:A\to _f B\to _g C\to 0$ is right-exact, i.e., $\mbox{im}...

I need help to resolve an apparent contradiction between part of a Proposition proved by Paul Bland in his book "Rings and Their Modules" and an Example provided by Joseph Rotman in his book "An Introduction to Homological Algebra" (Second Edition). One element of Bland's Proposition 3.2.7 is...

Homework Statement A 25 year old programme for building new houses began in Core Town in the year 1986 and finished in 2010. The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in 2000 and 108 in 2010, find the number of houses built in 1986...

Hey! :o Let $R$ be a commutative ring with unit. We have that if the sequences $0\rightarrow A\rightarrow B\overset{f}{\rightarrow}C\rightarrow 0$ and $0\rightarrow C\overset{g}{\rightarrow}D\rightarrow E\rightarrow 0$ are exact, then the sequence $0\rightarrow B\overset{gf}{\rightarrow}...

Homework Statement http://imgur.com/12LbqWL Part b Homework EquationsThe Attempt at a Solution Since it says the first four terms, not nonzero, the first four terms would be 0-(1/3-0)+2/9(x-2)-1/9(x-2)^2 I'm confused when it says I need to find these for x=2... Do I just plug in x=2 now and...

Hi everybody! I'm currently preparing a math exam, and I'd like to clear up a few points I find obscure about limits of sequences, my goal being to more or less determine a method to solve them quickly during the exam. Hopefully someone can help me here :) I'll number the questions so that it's...

Homework Statement \sum\limits_{n=1}^{\infty}\frac{n-1}{(n+2)(n+3)} Homework Equations S=\sum\limits_{n=1}^{\infty}a_n (1) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\gt 1\rightarrow S\ is\ divergent (2) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\lt 1\rightarrow S\ is\...

Homework Statement Determine which of the sequences converge or diverge. Find the limit of the convergent sequences. 1) {asubn}= [((n^2) + (-1)^n)] / [(4n^2)] Homework Equations [/B] a1=first term, a2=second term...an= nth term The Attempt at a Solution a) So I found the first couple of...

Homework Statement I have to determine whether or not the following sequence is convergent, and if it is convergent, I have to find the limit. an = (-2)n / (n!) In solving this problem, I am not allowed to use any form or variation of the Ratio Test. 2. The attempt at a solution I was...

I think this be Analysis, I Need some kind of convergence theorem for integrals taken over sequences of sets, know one? Example, a double integral taken over sets such that x^(2n)+y^(2n)<=1 with some integrand. I'd be interested in when the limit of the integral over the sequence of sets is...

Homework Statement [/B] Simplify: \frac{5\cdot 8\cdot 11 \cdots (3i+2)}{2\cdot 5 \cdot 8 \cdots (3i-1)} Homework EquationsThe Attempt at a Solution I realize the numerator and denominator terms cancel besides the 2, however I'm struggling to write this in a proper form. Only just started...

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...

I need the math tools to understand and analyze sequences and their convergence. I know for example that the fibonacci series can be rewritten such that we can calculate for example nr 153 without knowledge of previous numbers. What math subjects is needed to take care of more complicated...

Hello evry body let be $(u_{n}) \in \mathbb{C}^{\matbb{N}}$ with $u_{n}^{2} \rightarrow 1$ and $\forall n \in \mathbb{N} (u_{n+1) - u_{n}) < 1$. Why does this sequences converge please? Thank you in advance and have a nice afternoon:oldbiggrin:.