Geometric Definition and 813 Threads

Homework Statement Let {x_n} be a sequence. and let r be a real number 0<r<1. Suppose |x_(n+1) - x_n|<=r|x_n -x_(n-1)| for all n>1. Prove that {x_n} is Cauchy and hence convergent. Homework Equations if |r|<1 then the sequence \sum r^k from k=0 to n converges to 1/(1-r) The...

Homework Statement Suppose RX(t) = E[(1 − tX)−1] is called the geometric generating function of X. Suppose the random variable Y has a uniform distribution on (0, 1); ie fY (y) = 1 for 0 < y < 1. Determine the geometric generating function of Y . Homework Equations The Attempt at...

Hello, I have recently started to study some Geometric Algebra. I was wondering how should I interpret complex-vectors in \mathcal{C}^n in the framework of Geometric Algebra. I understand already that a complex-scalar should be interpreted as an entity of the kind: z = x + y (\textbf{e}_1...

what is the nth partial sum of 1-x+x^2-x^3+.. i don't understand why i can't do this? i have \sum_{k=1}^{n} ar^k=\frac{a(r^{n+1}-r)}{r-1} ok but then when i sub in i get messed up i get \frac{(-x)^{n+1}+x}{-(1+x)} is there any way of simplifying this? also when i set n=1 i get that...

Is it possible to find the partial sum equation for (2^m - 1)/3^m, from m=0 to m=n-1? I know that I'm supposed to rearrange the expression into the format ar^m, so the exponent m must only be on the value r, and not on the constant a. So far the farthest I've gotten is to rearrange it into...

Homework Statement In each case give a geometric description of the cosets of H in G. a) G=R*, H=R+ b) G=C*, H=R Homework Equations The Attempt at a Solution I have no idea about geometric description...

Using the formula for the sum of geometric series, show that the values of p(n) sum to 1 p(n)=(1 - \alpha)^n \alpha My attempt: \alpha \sum^\infty_{{\bf n=0}} (1- \alpha)^n I am not sure where to go from here. Any help to show this is true!

Homework Statement If A = [ -4e^4t sin(9t) -4e^5t cos (9t) ] [ 4e^4t cos (9t) -4te^5t sin (9t) ] then A^-1 = [ ___ ___ ] [ ___ ___ ] Homework Equations how do you deal with the exponents? The Attempt at a Solution Put into RREF, and then see what the inverse is?

Homework Statement actually got two questions but both are related so put them in the same place the question asks Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. Inf 1.) E 6(0.9)^(n-1) n=1 Inf (-3)^(n-1)...

Recently I discovered geometric algebra which looks very exciting. I was wondering if there is any connection between geometric algebra and differential forms? I see that different research groups recommend the use of differential forms (http://www.ee.byu.edu/forms/forms-home.html" ), and...

Homework Statement Calculate the angle of incidence for an angle of refraction of 10° for a) Diamond (2.42) to air θ2=Angle of refraction=10° n2=Index of refraction on refractive medium=1 n1=index of refraction on incident medium=2.42 θ1=? Homework Equations n1sinθ1=n2sinθ2...

Homework Statement Assume that each of your calls to a popular radio station has a probability of 0.02 of connecting, that is, of not obtaining a busy signal. Assume that your calls are independent. What is the probability that it requires more than five calls for you to connect...

Homework Statement http://img16.imageshack.us/img16/2327/nummer1.jpg Homework Equations Sn=(u1(rn-1))/(r-1) The Attempt at a Solution I think i need to use the equation for geometric series(above). Or do i use the arithmetic furmula since ln(a/b)=ln(a)-ln(b). I think i am a bit...

I'm trying to study geometric algebra using Artin's book and am having some difficulty with what degenerate symmetric bilinear forms would be like. Does someone know of an example and brief explanation. Also, the opposite being "nondegenerate nonsymmetric bilinear form" would help me out. If I...

I'm trying to build up enough understanding to work through some GR on my own, but I'm horribly confused by some of the math concepts. So terribly so, that I'm not even sure how to ask my questions. Please bear with me. Lets work in a 2D plane. Assume I have a vector u which I can write out...

Homework Statement Define the following terms: a) Midpoint M of segment AB. b) Perpendicular bisector of a segment AB (you may use the term "midpoint" since ypu have just defined it). c) Ray BDbisects angle ABC (given that point D is between A and C) d) Points A, B, and C are collinear...

Homework Statement I'm doing a report for a physics lab experiment where we are calculating the radius of the Earth by measuring the time it takes to see the sunset from the base of a cliff looking out into the pacific ocean till when it sets in relation to an observer at the top of the...

Homework Statement Draw and name all of the structural isomers of hydrocarbons with the formula C6H12 (cyclohexane). 2. Relevant information This is "only" a grade 11 Chemistry Assignment, therefore I only really need the basics when it comes to isomers- not some of the more complex...

We were discussing in an https://www.physicsforums.com/showthread.php?t=284322" that pinholes work like lenses, in that they form inverted real images. The other day, a friend told me, that pinholes even do bend the light rays like a lens do. I can - as being nearsighted - actually use a pinhole...

Three questions: 1) Anyone have a pedagogic answer to why the image formed by a thin lens is inverted (i.e. upside-down)? I realize that to get a focused image, the lens has to converge the rays one way or another, so then, eventually, the rays have to cross the optic axis, which in turn...

The Fibonacci numbers seem intimately connected with geometry. Prime numbers appear to avoid geometrics, however. Can you give some counterexamples of this latter statement?

Hi, I am trying to understand what Geometric Progression Weighted Average (GPWA) is in the context of the calculation of the US Dollar Index (USDX). I understand what a weighted average is but don’t understand what a GPWA is and when one should use it. The following equation shows how the...

1. Sum the Geometric Series 1-x+x2-x3+x4 and hence simplify the function [f(x)]4 = 1 - x5 1-x+x2-x3+x4 Homework Equations 3. Not sure I quite get understand this properly, as my attempt doesn't seem quite right. Basically I've gotten...

Homework Statement A Mooney rhomb is a quadrilateral prism that converts linear light to circular polarized light. The question is here...

Homework Statement (2+x)+(2+x)^2+(2+x)^3 + ... Homework Equations The Attempt at a Solution Ive found that the l r l < 1 the r of this equation is (2 + x) so we have -1 < 2 + x < 1 The values of x where the series coverges is -3 < x < -1 Is this correct...

If given the values -1, 5, 2 in this sequence, what would be the missing term to make this a geometric series? Also, what would the sum of this geometric series be?

Homework Statement 1+(x+1)+(x+1)^2+(x+1)^3 + ... if lx+1l < 1 Homework Equations Sn=a/1-r The Attempt at a Solution My attempt: so I have a = x+1 and r = x+1 from there i get x+1/1-(x+1) which is x+1/1-x-1 from there x+1/-x multiply by the reciprocal my...

There is a still-water lake and air interface. Light travels from the water to the air so that the incident angle is also a critical angle, making it so that the light runs along the surface of the water. Considering that this ray is reversible (air to water), a fish looking up at the surface...

Homework Statement Hi! I just used MATLAB to find the eigenvalues and eigenvectors of A=[0 -1; 1 0] I obtained the eigenvalues of 0 +/- i and eigenvectors of v(1) = [ 0.7071; 0 - 0.7071i] and v(2) = [ 0.7071; 0 + 0.7071i] Homework Equations I'm having trouble interpreting these results in...

Homework Statement The first 3 of a geometric progression are as follows: 2^{n}, 2^{n+1}, 2^{n+2} Find n Homework Equations For this kind of question, the only possibly helpful equation I can think of would be: T_{1}=a The Attempt at a Solution The problem I have is that at first...

Homework Statement Suppose A and B are similar matrices, and that (mu) is an eigenvalue of A. We know that (mu) is also an eigenvalue of B, with the same algebraic multiplicity(proved in class) Suppose that g is the geometric multiplicity of (mu), as an eigenvalue of B. Show that (mu) has...

Five positive integers P, Q, R, S and T, with P< Q < R <S < T, are such that: (i) P, Q and R (in this order) are in arithmetic progression, and: (ii) Q, R and S (in his order) are in geometric progression, and: (iii) R, S and T (in this order) are in harmonic progression. (I) Determine...

Ax = U \Sigma V^T x (A is an m by n matrix) I understand the first two steps, 1) V^T takes x and expresses it in a new basis in R^n (since x is already in R^n, this is simply a rotation) 2) \Sigma takes the result of (1) and stretches it The third step is where I'm a bit...

Vectors A and B each lie in the x-y plane. The magnitude of A + B equals the magnitude of A - B. Find the geometric relation between vectors A and B. (Hint: Express the vectors in unit-vector notation. Use your knowledge of dot products.) I really don't even know what this problem is asking...

The question from my textbook was "Give a complete proof that the largest triangle that can be inscribed in a circle is an equilateral triangle ", largest meaning that with the greatest area. I found a proof that finds the area of the general triangle in terms of the angles, and through...

I'm having difficulty understanding the geometry of a spacetime split as it applies to geometric algebra. I understand vectors, bivectors, and trivectors and I understand geometric products and wedge products. My impression is that a spacetime split let's you decompose an electromagnetic field...

In words, the sum of a geometric sequence can be written out to say "the first term divided by (1 minus the common ratio)". Does the first term also apply when the series starts with some other number n other than 1 (like 2 or 3, etc)? In other words, the first term is when n = some other number...

I've been stuck with this problem: An ellipse can be defined as 1) locus of points for which is constant the sum of the distances from two fixed points (foci) 2) locus of points for which is constant the ratio between the distances from a fixed point (focus) and a fixed line (directrix)...

Homework Statement Does the series from n=1 to infinity of (2)/(n^2-1) converge or diverge? If it converges, find the sum. Homework Equations The Attempt at a Solution I can see right away that the series converges by a limit comparison test by looking at the series. However, to find the sum...

Hi, I'm having trouble finding the sequence's total sum from a formula concerning Geometric Series. I've been using a calculator to find and manually input all of the terms into a table in Microsoft Excel and adding them all up at the end. The formula that I was given was...

Hi guys, In a geometric progression, the second term exceeds the first term by 20 and the fourth term exceeds the second term by 15. Find the possible values of the first term. ar - a = 20 ar3 - ar = 15 ar = 20 + a (20 + a)3 - (20 + a) = 15 But this method seem to be wrong. Any help?

Homework Statement On an exam question, although I can not remember the details, it gave us a table that looks similar to the table below. http://img403.imageshack.us/img403/9703/geoseqpr4.png [/URL] The question tells us that the graph is a geometric sequence, and says to use a formula to...

http://img117.imageshack.us/img117/5258/w1vg1.th.jpg http://img84.imageshack.us/img84/3151/w2px1.th.jpg See above files (one is the question and one is the answer) I can to the whole question, other than the last part - for part (f), why are we concerned with the sum to infinity of...

What the heck? The minimum monthly payment for a credit card is the larger of $5 or 1/25 of the outstanding balance. If the balance is less than $5, then the entire balance is due. If you make only the minimum payment each month, how long will it take to pay off a balance of $200? Clearly...

Homework Statement A mortgage is taken out for £80,000. It is to be paid by annual instalments of 5000with the first payment being made at the end of the first year that the mortgage was taken out. Interest of 4% is then charged on any outstanding debt. Find the total time taken to pay off the...

[SOLVED] Geometric Algebra: Signs of electromagnetic field tensor components? Here's a question that may look like an E&M question, but is really just a geometric algebra question. In particular, I've got a sign off by 1 somewhere I think and I wonder if somebody can spot it. PF isn't...

Hey, (sin A + sin B + sin C)/3 >= \sqrt[3]{}(sin A*sin B*sin C) I know this is true by Arithmetic mean always greater than geometric mean... but is there any other way of proving this?

I'd like the geometric meaning of the Mean Value Theorem. Say for instance I had a function of velocity that varied as t^{3} + 3t^{2} + 3t +1. I consider the interval [0,4]. So by MVT, I have a number c in [0,4] such that f'(c)(4) = f(4) - f(0). What does that mean? That there is an...

[SOLVED] geometric algebra: longitude and latitude rotor ordering? Was playing around with what is probably traditionally a spherical trig type problem using geometric algebra (locate satellite position using angle measurements from two well separated points). Origin of the problem was just me...

Homework Statement If a plane mirror is hung at an angle of 30 degrees from the vertical toward the viewer. what is the minimum length that will allow the viewer to see the full body image if the line of vision can not be depressed below the horizontal. you may assume that the eyes are at the...