Converging Definition and 190 Threads

Homework Statement Prove: If {Fn} converges to F on [a,b] and Fn is nondecreasing for each n\in N, then F is nondecreasing. Homework Equations n/aThe Attempt at a Solution First, it doesn't say if Fn converges pointwise or uniformly, so I'm not entirely sure how to deal with that. Just prove...

Homework Statement There is an object 20 cm from a converging lens with a 10 cm focal length. Homework Equations Is the image upright or inverted? Calculate the image distance. Calculate the image height. The Attempt at a Solution I just had this on a college test in a...

Im not sure how to do the part in red, do i use the comparison test?

Stuck on part iii, I am not sure what I've done wrong as my answer seems correct! any help wil be appreciated. And err, I can't seem to do part (iv) and (v), any help will be appreciated, thanks!

Prove that sum 1/2^(n+1)*n/(n+1), from n=0 to infinity, converges to 1 - log(2), where log stands for the natural logarithm. I know that the Taylor series for log(x) about x=1 is sum (-1)^(n+1)*(x-1)^n/n, but I don't see how these two statements are consistent. Thanks for any pointers!

Homework Statement (a)How far from a 50.0-mm-focal-length lens must an object beplaced if its image is to be magnified 2.00 x and be real? (b)What if the image is to be virtual and magnified 2.00x? Homework Equations m = -di/do 1/do+ 1/di = 1/f The Attempt at a Solution m...

Prove a lens must always be converging if... Consider a single lens with the object on the left side. Prove that for an upright, enlarged image that the lens must always be converging. I'm assuming my professor wants this to be mathematically proven, but I have no idea how to do so...HELP! I...

How can converging lens for sound waves be made? I got confused because i don't which materials should be used to make such lens. The lens if kept in air cannot be made of a solid material as sound travels faster in solid than in air and thus sound would not be converged. I have read that...

Hi, Have a question about images from a converging lens. I assume that the brightness of the (real) image projected on a screen depends upon the amount of light rays going through the lens - more rays -> brighter. So a larger lens should bring about a brighter image. Ok, now let's go the...

Homework Statement determine whether \sum n! / (n+2)! converges or diverges. if it converges, give the limit. The Attempt at a Solution i can see that this converges and the limit is 1/2 but i don't know how to mathematically show it... how can i factorise out the 1/2? so i have...

I looked through some tutorials to find intervals of divergence and tests for divergence... My series: [PLAIN]http://img843.imageshack.us/img843/4193/51453212.jpg a and x are constants... I did the ratio test and i get \rho=1, so i tried to apply the limit test to see if an is zero or does...

Venus, Mars and Saturn are converging on the Constellation Leo at the end of the month. I would like to find the Right Ascension and Declination of these planets given the obliquity, the true anomaly, the radius from the sun and the inclination.

Homework Statement Can a converging lens in air become a diverging lens in a different refractive medium? Homework Equations n1sin(θ1)=n2sin(θ2) The Attempt at a Solution I was thinking that it would not matter what medium the lens was in, as light would still hit the lens the...

If a Markov transition matrix's columns add up to 1, the matrix is a normal Markov transition matrix. Since it is normal, the Markov process must converge to a steady-state vector. Is this correct?

Homework Statement Starting with a real object, which of the following statements are TRUE about the image formed by a single lens? (Choose ALL correct answers, e.g. B, AC, CDF.) A) For a converging lens an object has to be placed between the focal length and the lens in order to form a...

I'm in the middle of studying for the MCAT, and just want to get all the concepts instead of memorizing how to solve problems. I'm having a hard time understanding converging lenses. From what I understand, converging lenses create real images that are inverted. This property of inversion...

Homework Statement A camper is trying to start a fire by focusing sunlight onto a piece of paper. The diameter of the sun is 1.39e9 m, and its mean distance from the Earth is 1.50e11 m. The camper is using a converging lens whose focal length is 10.8 cm. (a) What is the area of the sun's...

Homework Statement a1=0 an+1=Sqrt(1+2an) n=1,2,3... show that an converges and find the limit Homework Equations The Attempt at a Solution I have seen something like this before but i can't remember how to solve this. well i think i get that it becomes...

Homework Statement Solve 1 + 1/8 + 1/27 + 1/64... Homework Equations Stuck here. The Attempt at a Solution I know the series is infinitetly adding 1/n^3. I know it should converge. How do I solve as lim x--> infinity. Teacher never gave us any info, he just said use number sense.

Question: Is it possible for a given lens to act as converging lens in one medium and a diverging lens in another? What i was thinking: Yes. Let say we put the lens underwater. It slows the light down and turns the image upside down as it passes through the converging lens, turning it...

Homework Statement Homework Equations N/A The Attempt at a Solution Actually I read over this question for about 10 times already. But, I am not sure how to start. I know that I have to construct a subsequence one term at a time and show that it converges to L. But all the different...

Determine if the series n=2 to inf. of 1/ln(n) converges or diverges Ok so first I tried the limit test (the simple one) and found that it was 0 which was not helpful at all. Then I tried the integral test. It came out to be (integral)1/ln(n)=n/ln(n) + n/(ln(n))^2 + 2(integral from 2 to...

Homework Statement Hi all, I've been banging my head at this one for a while so any help is appreciated! Consider the series, \frac{1}{n}\sum_{k = 1}^{n} \frac{1}{k^{\delta}} where \delta \in (0,1). My teacher tells me that this series converges to zero but I am unable to prove that it...

Homework Statement Here is the layout of 2 lenses: -----------------(1)------------(2)-------------I-------------- Two thin lenses, numbered 1 and 2, with the intermediate image "I" is to the right of one of the lenses. The distance between the lenses is lettered d. Both lenses have...

A romantic, lighted candle is placed 39 cm in front of a diverging lens. The light passes through the diverging lens and on to a converging lens of focal length 10 cm that is 5 cm from the diverging lens. The final image is real, inverted, and 21 cm beyond the converging lens. Find the focal...

Homework Statement Assume that (an) is a bounded (but not necessarily convergent) sequence, and that the sequence (bn) converges to 0. Prove that the sequence (anbn) converges to zero. Homework Equations The Attempt at a Solution Assume that an is a bounded sequence and bn...

two converging mirrors A & B, having focal lengths 12 cm and 4 cm respectively, are facing each other and 36 cm apart on a common principal axis. A plastic sword is placed vertically upward in front of mirror A at a distance of 18 cm. Locate and describe the nature, orientation and size of the...

Homework Statement If (p_n/q_n)_{n \in \mathbb{N}} is a sequence of rationals in lowest terms converging to the irrational number x > 0, then \lim_{n \to \infty} q_n = +\infty. Homework Equations The Attempt at a Solution I'm quite clueless about this. But heuristically, I...

Homework Statement The figure illustrates qualitatively a converging lens focusing monochromatic rays from A to B. The vertical and horizontal scales of the figure are not the same. The top ray leaves A at an angle of 2.38 deg above the optic axis and leaves B at an angle of 1.25 deg below the...

Does there exist a converging uncountable sum of strictly positive reals?

Homework Statement In one of my problems, I'm asked to use integral test in order to determine whether the following function is converging or diverging: (summation from n=1 to inf) 1/(n^2-4n-5) I was just wondering how do we know if function is positive, continuous and decreasing because when...

Homework Statement Diverging lenses produce a "virtual" image, which can be seen but can't be projected, and converging lenses produce a "real" image which can be projected but not seen. How come we can see the image produced by a converging lens, which is supposed to be a real image...

A converging lens with f=33.7 cm yields an image with magnification M=−2. Find the object distance. Give your answer ... * in terms of centimeters. Do not include units in your answer. They will be assumed to be cm...

1. When an object 5.0 cm tall is placed 12 cm from a converging lens, an image is formed on the same side of the lens as the object but the image is 61 cm away from the lens. What is the focal length of the lens? 2. 1/f = 1/do + 1/di 3. 1/f = 1/do + 1/di = 1/12 + 1/61...

Homework Statement A laser beam passes through a converging lens with a focal length of 19cm. At what distance past the lens has the laser beam's intensity increased by a factor of 6? Homework Equations I=\frac{c\epsilon_0 E_0^2}{2} ?? The Attempt at a Solution To be honest, I'm...

Homework Statement Discuss the (absolute) convergence or divergence of hthe series whose nth term is: (ln(n))^(-ln(n)) Homework Equations The Attempt at a Solution I want to use the integral rule. First i show that it is decreasing by differentiating the function (and realising...

Homework Statement A converging lens has a focal length f, and a diverging lens has a focal length –f, which has the same magnitude as the converging lens. They are separated by a distance D which is greater than f, as shown. Parallel light enters from the left. Will the light be brought...

Homework Statement Show that the following series converges: Homework Equations Sum of (from n=1 to infinity) of [3^n + 4^n] / [3^n + 5^n] The Attempt at a Solution Some help on this question would be much appreciated as i really don't know how to start it. Thanks

Homework Statement Show that the following series converges: Homework Equations Sum of n=1 to n=infinity of [(n+1)/(n^2 +1)]^2 The Attempt at a Solution I thought about using the ratio test, so if the limit as n tends to infinity of a(n+1) / a(n) < 1, then a(n) - the series - would...

hi :) Quick questions: How can I write a series converging into the limit of pai ? Prove that the limit of the series zz (n+2) / (2n² +1) zz is not 1/2 when n->infinity (and bigger than 1) I came to the equation: zz |(-2n² + 2n +3) / (2n² +1) | >= ε zz (*)And I made the left...

Homework Statement There are 2 lenses separated by 5 cm. Very far to the left of the first lens is an object (the distance of the object from lens one is NOT given). The focal length of lens 1 is 20 cm and the focal length of lens 2 is 10 cm. Where is the image due to the first lens? State the...

Homework Statement Suppose that the sequence {an}converges. Show that the sequence {an} is bounded. The Attempt at a Solution Since the sequence converges, for every delta>0, there must exist a number N such that for every n>=N, |an - x|< delta. Therefore, for n>=N, -delta+x < an < delta...

Homework Statement A 1.5 cm tall object is placed 0.50 m to the left of a diverging lens with a focal length of 0.20 m. A converging lens with a focal length of 0.17 m is located 0.08 m to the right of the diverging lens. a)What is the location of the final image with respect to the object...

Homework Statement Prove or give a counterexample: If {an} and {an + bn} are convergent sequences, then {bn} is a convergent sequence. 2. The attempt at a solution Ok I couldn't think of any counterexamples, so I tried to prove it using delta epsilon definitions: |an - L| < E |an...

Homework Statement My task was to determine the focal distance and the magnification of a converging lens. I've done everything as stated in the procedure. The problem that i have is that when i am calculating the magnification in two differents ways i.e M1=b/a and M2=h'/h where a=distance...

Homework Statement Let H be a Hilbert space. Prove that if \left\{ x _{n} \right\} is a sequence such that lim_{n\rightarrow\infty}\left\langle x_{n},y\right\rangle exists for all y\in H, then there exists x\in H such that lim_{n\rightarrow\infty} \left\langle x_{n},y\right\rangle =...

Homework Statement For each of the following object positions in front of a converging lens, suggest at least one practical application. (E.g. overhead projector, camera etc.) a) Between the lens and F' b) At F' c) Between F' and 2F' d) At 2F' e) Beyond 2F' The Attempt at a...

Homework Statement a) A converging lens with a focal length of 15cm has an object 5cm tall placed 40cm to the left of it. To the right, 30 cm away, is a converging lens of focal length 10cm. use the thin lens and magnification formulae to locate the finale image. Find its magnification. b)...

Homework Statement Consider the sequence \left\{ x_{n} \right\}. Then x_{n} is convergent and \lim x_{n}=a if and only if, for every non-trivial convergent subsequence, x_{n_{i}}, of x_{n}, \lim x_{n_{i}}=a. Homework Equations The definition of the limit of a series: \lim {x_{n}} = a...

Homework Statement Lets say that I have some sequence (a_n) which converges to 0 at infinity and that for all n a_{n+1} < a_n but the sequence (a_n) diverges. Now I know that the series (cos(n) a_n) converges but can I use the following argument to prove that |cos(n) a_n| doesn't...