Convergent Definition and 330 Threads

From my physical problem, I ended up having a sum that looks like the following. S_N(\omega) = \sum_{q = 1}^{N-1} \left(1 - \frac{q}{N}\right) \exp{\left(-\frac{q^2\sigma^2}{2}\right)} \cos{\left(\left(\mu - \omega\right)q\right)} I want to know what is the sum when N \to \infty. Here...

So far this is what I have. Proof: Let p1, p2, p3 be a non-decreasing sequence. Assume that not all points of the sequence p1,p2,p3,... are equal. If the sequence p1,p2,p3,... converges to x then for every open interval S containing x there is a positive integer N s.t. if n is a positive integer...

Let's us look at the first implication (I will post the reverse implication once this proof has been verified). We have to prove if there is a subsequence of ##(s_n)## converging to ##t##, then there are infinitely many elements of ##(s_n)## lying within ##\epsilon## of ##t##, for any...

Dear Everybody, I have a quick question about the \M\ in this proof: Suppose \b_n\ is in \\mathbb{R}\ such that \lim b_n=3\. Then, there is an \ N\in \mathbb{N}\ such that for all \n\geq\, we have \|b_n-3|<1\. Let M1=4 and note that for n\geq N, we have |b_n|=|b_n-3+3|\leq |b_n-3|+|3|<1+3=M1...

Let ##\epsilon>0##. Then there is an integer ##N>0## with the property that for any integer ##n\geq N##, ##|a_n-A|<\epsilon##, where ##A\in\mathbb{R}##. If for all positive integers ##n##, it is the case that ##|a_n-A|<\epsilon##, then the following must hold: \begin{eqnarray}...

Hello i have problems with this exersice Let $$\{X_{\alpha}\}_{\alpha \in I}$$ a collection of topological spaces and $$X=\prod_{\alpha \in I}X_{\alpha}$$ the product space. Let $$p_{\alpha}:X\rightarrow X_{\alpha}$$, $$\alpha\in I$$, be the canonical projections a)Prove that a sequence...

Ok I am trying to brush up my real analysis skills so that I can study some topology and measure theory at some point. I found this theorem in my notes, that is proven by using proof by contradiction. However, I have a hard time understanding what the contradiction really is... Here is the...

Hi, I was recently reading about convergent-divergent nozzles and was wondering about how boundary layers grow in them. Question: How does a boundary layer grow in a convergent duct in subsonic flow? How does this compare to the growth of a boundary layer in a divergent duct in subsonic flow...

Mary Boas attempts to explain this by pointing out that the situation cannot arise because charges will have to be placed individually, and in an order, and that order would represent the order we sum in. That at any point the unplaced infinite charges would form an infinite divergent series...

##\sum_n \frac{1}{n^c}## converges for ##c\gt 1##. Is there an expression for the value of the sum as a function of ##c##?

I want to present two scenarios here. First, there is a convergent nozzle shaped structure having a fan/blower fitted inside. In this case, the fan/blower is fitted at the throat of the nozzle. The inlet to throat ratio isn't important here. And the second scenario, the same nozzle is used but...

find out for what values of p > 0 this integral is convergent ##\displaystyle{\int_0^\infty x^{p-1}e^{-x}\,dx}\;## so i broke them up to 2 integrals one from 0 to 1 and the other from 1 to ∞ and use the limit convergence test. but i found out that there are no vaules of p that makes both of...

Use the Ratio Test to determine whether the series is convergent or divergent $$\sum_{n=1}^{\infty}\dfrac{(-2)^n}{n^2}$$ If $\displaystyle\lim_{n \to \infty} \left|\dfrac{a_{n+1}}{a_n}\right|=L>1 \textit{ or } \left|\dfrac{a_{n+1}}{a_n}\right|=\infty...

Let A ⊆ R, let f : A → R, and suppose that (a,∞) ⊆ A for some a ∈ R. Then the following statements are equivalent: i) limx→∞ f(x) = L ii) For every sequence (xn) in A ∩ (a,∞) such that lim(xn) = ∞, the sequence (f(xn)) converges to L. Not even sure how to begin this one, other than the fact...

This time, i have a question that came to my mind a few days ago. There is an ideal nozzle having inlet to throat ratio of 8:1. Air will enter the nozzle at around 84 m/s velocity. it can be easily understood that the velocity will become sonic much ahead of the throat. And I am wondering what...

I want to know what's the most efficient type convergent nozzle available now at present. I means what's the shape that is most efficient for a convergent nozzle. By efficiency, I want to mean that the inlet velocity to throat velocity ratio will be as close as possible to the throat area to...

Hello! (Wave) Let $(a_n), (b_n), (c_n)$ sequences such that $(a_n), (c_n)$ are bounded and $a_n \leq b_n \leq c_n$ for each $n=1,2, \dots$ I want to show that $(b_n)$ has a convergent subsequence. I have thought the following: Since $(a_n), (c_n)$ are bounded, $\exists m_1, m_2 \in...

Homework Statement Prove that every convergent sequence has a monotone subsequence. Homework EquationsThe Attempt at a Solution Define ##L## to be the limit of ##(a_n)##. Then every ##\epsilon##-ball about L contains infinitely many points. Note that ##(L, \infty)## or ##(-\infty, L)## (or...

Homework Statement Prove or produce a counterexample: If ##\{a_n\}_{n=1}^\infty\subset \mathbb{R}## is convergent, then ## \min (\{a_n:n\in\mathbb{N}\} )## and ##\max (\{a_n:n\in\mathbb{N}\} ) ## both exist. Homework EquationsThe Attempt at a Solution I will produce a counterexample...

Let us assume we have a cylindrical wind tunnel having a 0.5 m diameter fed by an electric fan. The cross-sectional area of the wind tunnel would be A1 = (PI/4) D1^2 = 0.196349541 m2. Let us suppose the motor driving the fan has a power rating of 1,500 W. At this stage, let us assume that the...

Recently I have some conversation with an electrical engineer and from him I cam to know about a peculiar information. We were talking about what will happen if we put a convergent nozzle before a electricity driven fan/blower. I have concentrated what will happen at the throat and other...

Homework Statement Homework Equations - The Attempt at a Solution Here's my work : However , the correct answer is : Can anyone tell me where's my mistake ?

Hey! :o I want to show that the Gamma function converges and is continuous for $x>0$. I have done the following: The Gamma function is the integral \begin{equation*}\Gamma (x)=\int_0^{\infty}t^{x-1}e^{-t}\, dt\end{equation*} Let $x>0$. It holds that...

Homework Statement ##\sum_{n=1}^{\infty }1+(-1)^{n+1} i^{2n}## Is this series divergent or convergent? Homework Equations 3. The Attempt at a Solution [/B] I tried using the divergent test by taking the limit as ##n## approaches ##{\infty }##, but both ##i^{2n}## and ##(-1)^{n+1}## will...

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...

Homework Statement Determine if the series is convergent. Homework Equations ∞ ∑ (((2n^2 + 1)^2)*4^n)/(2(n!)) n=1[/B] The Attempt at a Solution I'n using the Ratio Test and have got as far as (4*(2(n+1)^2+1)^2)/((n+1)((2n^2+1)^2)). I know this series converges but I need to find the...

Homework Statement "Let ##\{a_n\}_{n=1}^\infty## be a bounded, non-monotonic sequence of real numbers. Prove that it contains a convergent subsequence." Homework Equations Monotone: "A sequence ##\{\alpha_n\}_{n=1}^\infty## is monotone if it is increasing or decreasing. In other words, if a...

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...

Homework Statement Two convergent lens are identical in focal length (f=10cm) and the oobject height (h0) is 3.5 cm. The distance between the two lenses is 30cm and the distance from the object to the first lens is 30cm. -Draw a diagram on the figure and show the image position (di) and size...

Homework Statement Assume that ##a_k > 0## and ##\sum_{k=0}^\infty a_n## converges. Then for every ##\epsilon > 0##, there exists a ##n \in Bbb{N}## such that ##\sum_{k=n+1}^\infty a_k < \epsilon##. Homework EquationsThe Attempt at a Solution Since the series converges, the sequence of...

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...

Poster warned that the homework template is not optional. Determine if they are convergent or divergent, If it converges find the sum: ∞ ∑ 3^(n-1) 2^n n=1 ∞ ∑ ln(1/n) n=1 ∞ ∑ tan^n ( π/6) n=1 I tried to find information on how to solve them but I couldn't, thanks for the help

Homework Statement I have a couple of series where I need to find out if they are convergent (absolute/conditional) or divergent. Σ(n3/3n Σk(2/3)k Σ√n/1+n2 Σ(-1)n+1*n/n^2+9 Homework Equations Comparison Test Ratio Test Alternating Series Test Divergence Test, etc The Attempt at a...

Homework Statement Consider the space ##([0, 1], d_1)## where ##d_1(x, y) = |x-y|##. Show that there exists a sequence ##(x_n)## in ##X## such that for every ##x \epsilon [0, 1]## there exists a subsequence ##(x_{n_k})## such that ##\lim{k\to\infty}\space x_{n_k} = x##. Homework Equations N/A...

Is it any argument structure not classified as a syllogism? where premises lead to conclusions which is another premise. It seems that the definition is that in a convergent argument all the premises are independent of each other and support the conclusion only. But how does one know? "I...

Homework Statement [/B] Hi Theorem attached and proof. I am stuck on 1) Where we get ##|g(z)|\geq |a_m|/2 ## comes from so ##a_{m}## is the first non-zero Fourier coeffient. So I think this term is ##< |a_m|r^{m}##, from ##r## the radius of the open set, but I don't know how to take care...

I have: $$\int_{1}^{2} \frac{1}{x lnx} \,dx$$ I can set $u = lnx$, therefore $du = \frac{1}{x} dx$ and $xdu = dx$. Plug that into the original equation: $$\int_{1}^{2} \frac{x}{x u} \,du$$ Or $$\int_{1}^{2} \frac{1}{ u} \,du$$ Therefore: $ln |u | + C$ and $ln |lnx | + C$ So I need to...

Homework Statement Homework Equations Ratio test. The Attempt at a Solution [/B] I guess I'm now uncertain how to check my interval of convergence (whether the interval contains -2 and 2)...I've been having troubles with this in all of the problems given to me. Do I substitute -2 back...

Homework Statement ∞ Σ (-1)n-1 n/n2 +4 n=1 Homework Equations lim |an+1/an| = L n→∞ bn+1≤bn lim bn = 0 n→∞ The Attempt at a Solution So I tried multiple things while attempting this solution and got inconsistent answers so I am thoroughly confused. My work is on the attached photo. I found that...

I have this: $$ \sum_{n = 1}^{\infty} \frac{n^n}{3^{1 + 3n}}$$ And I need to determine if it is convergent or divergent. I try the limit comparison test against: $$ \frac{1}{3^{1 + 3n}}$$. So I need to determine $$ \lim_{{n}\to{\infty}} \frac{3^{1 + 3n} \cdot n^n}{3^{1 + 3n}}$$ Or $$...

Question ∞ ∑ tan(1/n) n = 1 Does the infinite series diverge or converge? Equations If limn → ∞ ≠ 0 then the series is divergent Attempt I tried using the limit test with sin(1/n)/cos(1/n) as n approaches infinity which I solved as sin(0)/cos(0) = 0/1 = 0 This does not rule out anything and I...

I'm really confused about this test. Suppose we let f(n)=an and f(x) follows all the conditions. When you take the integral of f(x) and gives you some value. What are you supposed to conclude from this value?

It's a well known fact that convergent and/or convergent-divergent nozzles convert internal enthalpy into forward motion i.e. dynamic pressure. But, enthalpy means both internal heat and the pressure-volume. I want to know whether the internal heat of a fluid can also be converted into forward...

Do (i), (ii) and (iii) apply to conditionally convergent series as well? I feel like they don't. But the book seems to say that they do because it doesn't "state otherwise".

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 This is a problem from Haim Brezis's functional analysis book. Homework EquationsThe Attempt at a Solution I'm assuming (e)n is the vectors like (e)1 = (1,0,0), (e)2=(0,1,0) and so on. We know every hilbert space has an orthonormal basis. I also need to know the...

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

Sorry for the bad English , do not speak the language very well. I posted this to know if the statement or " hypothesis " is correct . thank you very much =D. First Image:https://gyazo.com/7248311481c1273491db7d3608a5c48e Second Image:https://gyazo.com/d8fc52d0c99e0094a6a6fa7d0e5273b6 Third...

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...