Converging Definition and 190 Threads

Homework Statement So I solved this problem ∑ (-1)^n (x+5)^n by finding the sum of the series to be 1/(x+6), which was correct. I am stuck on the second part of the problem asking for interval notation. It says: Determine, in interval notation, the values of x for which the series converges...

Homework Statement Two converging lenses of the same focal length f are separated by distance 2f. The axis of the second lens is inclined at angle θ = 60º with respect to the axis of the first lens. A parallel paraxial beam of light is incident from left side of the lens. Then: (A)...

Hello. I would like to inquire as to how to deal with the said topic title. I'm trying to generate a VLE graph for ethylene oxide-water. While I know that EO will quickly vaporize since the boiling point of EO is quite low, I'm still trying to generate a VLE using SRK. So while the vapor...

Hi everyone, I was thinking about an optics question in a physics textbook. I think the solutions I saw in the solution manual may be wrong. The question is: When you do the calculations using 1/p + 1/q = 1/f you find that the lens first creates an image 400 cm to the...

Homework Statement 1.Where would you place an object to obtain an image as far away from the lens as possible? 2. Suppose you have an image focused on a slide screen. You moved the image towards the lens. Which way would you move the lens to keep the image in focus? Explain Homework...

Let's imagine a rocket that travels straight path through space, accelerating gradually (or being boosted by outside force). The speed of rocket is such that 1st light year takes 1/2 hours in rocket's own frame 2nd light year takes 1/4 hours 3rd light year takes 1/8 hours 4th light year...

Dear reader, I have written a paper which I require a reference for converging wave interference generating a caustic. When studying for my degree, I performed Young's double slit experiment with the addiction of a converging lens. I was trying to comprehend Afshar's...

How do I prove that given Xn converges almost surely to X, that f(Xn) will converge almost surely to f(X)?

Homework Statement Two identical converging lenses have focal lengths of 15 cm and are aligned on the same principal axis. They are separated by a distance of 120 cm. An object sits a distance of 21 cm to the left of the left lens. What is the total magnification of the two lenses...

Homework Statement An object is in front of a converging lens with a focal length of 11.0 cm. The lens produces a real image located 25.0 cm from the lens. If the lens is replaced by a new lens (in the same position) that has a focal length of 32.0 cm, what is the image distance of the...

Here is the problem at hand: \lim\limits_{n\to\infty} \frac{\sqrt{n}}{2^{2n}} \binom {2n} {n+\lfloor x\sqrt{n} \rfloor} = \frac{1}{\sqrt{\pi}} e^{-x^2} I've tried to evaluate the limit using the Stirling's formula, many things canceled out yet I got stuck. Thanks a lot for any help or...

Homework Statement Show for what real numbers p and q ## \sum \frac{1}{n^{p}ln^{q}(n)} ## diverges or converges. Homework Equations The Attempt at a Solution I am kind of lost because it seems that with both subscripts p and q there are a bunch of cases you have to work through...

Homework Statement Air flows through a symmetric converging-diverging nozzle where the cross-sectional area of the nozzle (in meters^2) varies accoding to the relationship: A(x) = 1.0 - 0.8X +.80X^(2) ; where X is in meters and the nozzle is 1 meter long. P2/P1 = .90 (there is...

Homework Statement A real object is placed 9.00 cm from a converging lens that has a focal length of 24.0 cm. The image is________. My classmate answered "14.4 cm from the lens, virtual, and enlarged," but apparently this is incorrect. I solved it out and chose the option "14.4 cm from the...

Homework Statement This is the given question. The blue rays represent the solution. http://img16.imageshack.us/img16/7237/lenstq.jpg Homework Equations None. The Attempt at a Solution The blue rays are essentially my solutions, but only up to the point where the rays are...

Optics -- magnification of a converging lens Homework Statement Dioptry of converging lens is $D=3$. What is magnification ##u##? Homework Equations The Attempt at a Solution ##\frac{1}{f}=D## - dioptry. \frac{1}{f}=\frac{1}{p}+\frac{1}{l} u=\frac{l}{p} l=up...

Homework Statement General question here: If I have a lens that looks like two concave lenses put together (I believe it is called a Meniscus), how do I conceptually tell whether the lens should be converging or diverging? I think I remember reading something saying that it depends upon which...

Homework Statement Let f be a 2π-periodic function (can be any periodic really, not only 2π), and let g be a smooth function. Then lim_{n\rightarrow∞}\int^{B}_{A} f(nx)g(x) converges to \frac{1}{2π}\int^{2π}_{0}f(x) The Attempt at a Solution So far, I've come up with somewhat of...

Does there exist a sequence of real nonzero numbers whose sum converges to 0? I would think there isn't, but I'm interested in people's opinions and arguments. For any nonzero m, a series of nonzero numbers whose sum converges to m can easily be constructed using the formula: \sum...

Homework Statement A certain lens focuses light from an object 2.90m away as an image 46.9cm on the other side of the lens. 1. What type of lens is it? (Converging or Diverging?) 2. What is the focal length? 3. Is the image real or virtual? Homework Equations 1/di + 1/do = 1/f...

Homework Statement I want to prove that if a sequence a[n] is cauchy then a[n] is a converging sequence Homework Equations What I know is: a[n] is bounded any subsequence is bounded there exists a monotone subsequence all monotone bounded sequences converge there exists a...

I come from a civil engineering background and I'm about to start a masters in nuclear energy. I have no experience in thermodynamics and I've been doing a bit of revision before the course commences to get me up to speed. I've been working my way through 15 revision questions and I only have a...

Homework Statement A converging lens with a focal length of 14.2 cm is used to inspect a coin. The lens forms a virtual image 30.1 cm away. What is the magnification? Homework Equations (1/Si) + (1/So) = (1/f) M = (hi/ho) = (-Si/So) The Attempt at a Solution 1/So = 1/f - 1/Si...

Homework Statement Suppose that a mathematically inclined child plays with a basket containing an infinite subset of integers (with some repetitions). If an integer k is present in the basket then there are initially |k| copies of it. The child pulls out the integers from the basket at...

Determine whether each of the following sequences converge or not. In each case present a formal explanation. If a sequence converges find the limit and if not determine whether there should be any converging subsequences, and if so find more than one converging subsequences. Xk+1 =...

Homework Statement A compound microscope consists of two converging lenses. Lens 1, called the objective, is a distance 61.0 mm from lens 2, called the eyepiece. The objective is placed in between the object and the eyepiece. The objective has a focal length of 15.0 mm, and the eyepiece has...

Homework Statement Which type of lens can have a magnification greater than 0? Which statement is most complete? A) Both, and both converging and diverging lens can have any m > 0. B) A diverging lens only. C) Both, but a converging lens has m > 1 while a diverging lens has m between 0...

Homework Statement There are two converging lenses. The object of the first lens (i.e. the lens on the left) is 17.5 cm to the left of the first lens, and the resulting image from the first lens is 23.33 cm to the right of the first lens. The second lens is 40 cm to the right of the first...

This is not homework but a problem I would like to learn to solve before my exam. Homework Statement https://dl.dropbox.com/u/23889576/Screenshots/01.png Homework Equations Series Formulae The Attempt at a Solution I went about trying to determine a sum formula for the fly's...

Homework Statement Question shows one arrow with 10 amperes heading in one direction, which splits into three different pathways. having two labels of 4 and 6 amperes. Homework Equations To be honest I don't know what equations I would use; the courseware really wasn't much help...

Probably a complete newb question and I know magnification is dependent on how severely the light is bent but what physical aspects cause the light to bend more/less? I was told it's the thickness so do thin or thick lenses bend light more severely? and which would result in greater...

I have the following exercise: Let s_n be a cauchy sequence in Q(rationals) not converging to 0. Show that there exists an e(epsilon) >0 and a natural number N such that either for all n>N, s_n > e or for all n>N, -s_n >e. I know that since Q is not complete, we cannot assume that there...

1. Imagine a positive point x not equal to zero. 2. Consider a randomly chosen point y with distance to zero less than x. 3. Let y=x. Repeat #2. 4. Is the sum of the y-values finite as y approaches zero?

Homework Statement The gamma function, which plays an important role in advanced applications, is defined for n\geq1 by \Gamma(n)=\int_0^{\infty} t^{n-1}e^{-t}dt (a) Show that the integral converges on n\geq1 (b) Show that \Gamma(n+1)=n\Gamma(n) (c) Show that \Gamma(n+1)=n! if n\geq1 is an...

Homework Statement The points a, b and c and mid way along a series of tubes arranged such: There is a tube of 4mm for 100 cm It splits into 3 tubes of diameter 1mm, length 20 cm It converges into a tube of diameter 6mm, length 100 cm The fluid flowing through the tubes is 3ml/sec...

Homework Statement A converging lens of focal length 7.740 cm is 19.7 cm to the left of a diverging lens of focal length -5.51 cm. A coin is placed 18.2 cm to the right of the diverging lens a. Find the location of the coin's final image. b. Find the magnification of the coin's...

Hi everyone, I need to show that \lim_{n\to\infty}P_n(x) = |x| uniformly on [-1,1] if we define P_0 = 0 & P_{n+1} = P_n(x) + \frac{x^2 - P_n^2(x)}{2} Rudin gives the following hint: use the identity |x| - P_{n+1}(x) = \left( |x| - P_n(x) \right) \left( 1 - \frac{|x|+P_n(x)}{2} \right) to show...

Probability integral not converging :( Homework Statement for the joint probability density function: f(x,y)= \begin{cases} y & 0 \leq x, \, y\leq1 \\ \frac{1}{4} (2-y) & 0\leq x\leq 1, \, 0\leq y\leq 2 \\ 0 & \text{elsewhere} \end{cases} find the following: a) f_1 (x) b) f_2...

Homework Statement Suppose that f_n \stackrel{P}{\rightarrow} f on S. Given \epsilon >0 define E_{\epsilon,n}=\{x\in S \mid |f_n(x)-f(x)|<\epsilon\}. Prove that \forall \epsilon >0 \cup_{n=1}^\infty E_{\epsilon,n}=S. Homework Equations Know about measurability, pointwise convergence...

Homework Statement Let f(x,y)=xe^{-xy} x \geq 0, y \geq 1 is this a probability density function? If not, find a constant that makes it a pdf. Homework Equations To be a pdf, we must have \int_1^\infty \int_0^\infty \! xe^{-xy} \, \mathrm{d} x \mathrm{d} y=1 The Attempt at a...

A converging meniscus lens has an index of refraction of 1.55 and radii of curvature for its surfaces of 6.00 cm and 11.0 cm. The concave surface is placed upward and filled with carbon tetrachloride which has n = 1.46. What is the focal length of the glass combination? So I tried using the...

Hey folks just wondering whether anyone here knew of a series for cos(x) that converges fairly quickly for values between 0 and 2*pi? The 'usual' series takes up to around the 10th term before it starts looking decent for cos(2*pi) for example.

Homework Statement a converging diverging nozzle is operating at sea level, it will be mounted on a glider that flies 26.8m/s. The entrance area is 1 m^2. Homework Equations what is the stagnation pressure and mach number of flow entering the nozzle? what should be throat and exit...

Homework Statement Two converging lenses, each of focal length 14.7 cm, are placed 39.7 cm apart, and an object is placed 29.3 cm in front of the first. How far from the first lens is the final image formed? What is the magnification of the system? Answer in units of cm. Homework...

Homework Statement A converging lens of focal length 0.246 m forms a virtual image of an object. The image appears to be .933 m from the lens on the same side as the object. What is the distance between the object and the lens? Homework Equations 1/f = 1/di + 1/do Since the image is...

Homework Statement An Object and a screen are located a fixed distance D apart. (A) Show that a converging lens of focal length F placed between the object and screen will form a real image on the screen for two positions that are separated by a distance d=(D(D-4F))^1/2 (B) Show that...

Given the sequence: if n=1, an = 2 if n>1, an+1 = 1/2(an + 3/an) prove that this sequence is decreasing im having trouble with recursively defined sequences. I know I am supposed to use induction in some way, but its not that straitforward with the 'double sequence' in the an+1...

Homework Statement Prove that if the sequence {An} converges to A, then the sequence {|An|} converges to |A|. And is the converse true? This is for my calculus class and it needs to be in proof format. Thank you! The Attempt at a Solution I'm totally lost, I was going to use ||x| -...

Dear friends can you show me please an example of a sequence of integrable functions fn:R->R converging to an integrable function f but *not* in the L1-norm, i.e. such that \Int \mid f_n -f\mid is not equal to 0? Thank u a lot

Homework Statement Give an example of a continuous, non-negative function f: [1, infinity) --> R such that if an = f(n) for each positive integer n, the series \sum an diverges, while the improper integral from 1 to infinity of f converges. Justify your answer. Homework Equations N/A...