Geometric Definition and 813 Threads

Hi. I've got a theoretical doubt: why is the geometric multiplicity more or equal to 1? Couldn't happen that the eigenspace is the null vector? Thanks!

Algorithms for infinite geometric series via long division?? I can't seem to find any algorithms for this on the internet easily. If I have a function of the form f(x)=\frac{a}{x+b} there should be an algorithm I can use to find some terms of the corresponding series \sum...

Hello, I am working through Clifford Algebra to Geometric Calculus, and supplementing with Hestenes' other books, as well as Geometric Algebra for Physicists. I'm not looking for advice on the books or learning materials (feel free to chime in if you have an opinion on the matter...

Geometrically, what is the difference between saying 'X is homotopic equivalent to Y' and 'X is homeomorphic to Y'? I know that a homeomorphism is a homotopy equivalence, but I can't seem to visualise the difference between them. It seems to me that both of these terms are about deforming spaces...

Basically, we have to prove that A X B is equal to ABsinθ Now a crucial step towards the proof is proving that (AXB).(AXB) is equal to (AB)2 - (A.B)2 After that it is fairly simple. But unfortunately, I can not prove the identity. I've tried expanding it into components but things are...

Phi ~ 1.618

Homework Statement I am after finding general geometric expressions for a quarter-circular spandrel that is split into two segments along either its domain or range (they are equal). I.e. Taking the geometry provided in the sketch below (Figure 1) I am after expressions for area and centroids...

I have a series presented to me that goes something like this- 1/1+1/2+2/4+3/8+5/16+8/32+... I am aware that it is a sum to inifinity problem and the common ratio of the bottom is 2 but I don't know what to do with the numerator.

I'm tying to solve a stochastic differential equation of stock price. The equation is dX = X(\mu dt + \sigma dW) where \mu, \sigma are constants and greater than zero. It is easy to show analytically that the expectation value to the solution is E[X(t)] = E[X(0)] e^{\mu t} Then I solved...

Homework Statement The Bank of Utopia offers an interest rate of 100% per annum with various options as to how the interest may be added. A man invests $1000 and considers the following options. Option A - Interest added annually at the end of the year. Option B - Interest of 50% credited...

Homework Statement A 'supa-ball' is dropped from a height of 1 metre onto a level table. It always rises to a height equal to 0.9 of the height from which it was dropped. How far does it travel in total until it stops bouncing? Homework Equations The Attempt at a Solution The...

Homework Statement Evaluate the problem: \sum_{k=1}^\infty \frac{3^{(k-1)}}{4^{(k+1)}} Homework Equations \displaystyle\lim_{n\rightarrow\infty} S_n = \frac{a}{1-r} The Attempt at a Solution I know that the limit of the partial sequence is what i need to help solve this, but can't...

Part c) I'm not quite sure what to do, I've found the det(U) is 2, but no idea what this actually shows to be honest, any help?

Let's say I have a series of 100 coin tosses, heads or tails. In fact (for my actual data) I don't know if subsequent trials are correlated or what the actual probabilities of getting heads or tails are. Nevertheless, I want to fit a geometric distribution, which gives me the distribution of the...

I'm struggling with trying to visualize the vector potential as in the identity: B = ∇⨯A For starters, how does A relate to, say, a uniform magnetic field, which is quite easy to visualize. Then, how about the magnetic field around a bar magnet -- where is A? Any help would be appreciated.

Homework Statement With a series like: pi^(n/2)*cos(n*pi) How am I meant to approach this? Do I use the Squeeze Theorem? Homework Equations The Attempt at a Solution

Homework Statement Write the first 5 terms of the geometric sequence 3, __ , 32x+1, __, __ Homework Equations tn=arn-1 The Attempt at a Solution tn=arn-1 32x+1=3r3-1 r2=32x+1 / 3 r=√32x+1 / √3 I'm stuck

Homework Statement Suppose the sequence (Sn) is defined as: |Sn+1-Sn|<2-n show that this is a cauchy sequence Homework Equations hint: prove the polygon identity such that d(Sn,Sm)≤d(Sn,Sn+1)+d(Sn+1,Sn+2)...+d(Sm-1,Sm) The Attempt at a Solution I have defined Sm and Sn and created the...

Homework Statement A ball is dropped from a 100 feet and has a 90% bounce recovery. How many bounces does it take for the ball to travel 1854.94320091 feet? Homework Equations -None- The Attempt at a Solution I know the ratio is .9 and the 'a one' value is 180, so I plugged those values in...

Homework Statement Different numbers x, y and z are the first three terms of a geometric progression with common ratio r, and also the first, second and fourth terms of an arithmetic progression. a. Find the value of r. b. Find which term of the arithmetic progression will next be equal to...

Homework Statement Find the first term in this geometric sequence that exceeds 500. 2, 4, 8, 16, ... Homework Equations Un = arn-1 The Attempt at a Solution a = 2, r = 2 Un = 2 x 2n-1 > 500 2 x (2n)(2-1) > 500 log22 x log22n + log22-1 > log2500 1 x n + (-1) > log2500 n - 1 >...

Hello, I'm currently working through Hestenes' and Sobczyk's book "Clifford Algebra to Geometric Calculus." It has been slow reading because of the many skipped steps in his derivations (I'm not saying that's a bad thing), but I am rather enjoying GA/GC so far. I work through all of the...

A man gets a credit card and buys something that charges exactly 800 dollars to the card. The APR on the card is 18 % compounded monthly, and the minimum payment is 15 dollars a month. What is the expression for A(n), the balance on the card after n months? (This should be a geometric series)...

Use geometric series to find the Laurent series for f (z) = z / (z - 1)(z - 2) in each annulus (a) Ann(1,0,1) (b) Ann(1,1,∞) Ann(a,r,R) a= center, r=smaller radius, R=larger radius Ann(1,0,1)=D(1,1)\{0} My attempt: f(z)= -1/(z-1) + 2/(z-2) geometric series: Σ[n=0 to inf] z^n - 1/2...

Homework Statement Ʃ(1 to infinity) (2/3)^(3n)Homework Equations For a geometric series, the series converges to a/1-r The Attempt at a Solution I'm really just confused about how to manipulate this so that it has a form of ar^n, especially since it starts at 1 rather than 0. I know that...

Hi! Probably I am just confused, but why for the exact solution of the geometric brownian motion dX_t = \mu X_t dt+\sigma X_t dW_t we have to apply Ito's lemma and manipulate the expression obtained with dlogX_t? Couldn't we directly use the espression dX_t / X_t = dlogX_t in the equation dX_t /...

Hi, I have the following equation: \gamma=\frac{1}{\frac{1}{N}\sum_{n=1}^N|\lambda_n|^{-2}} where lambdas are the eigenvalues of an N-by-N circulant matrix A. I used two properties to bound the above equation...

On this website http://www.albany.edu/~bd445/Eco_466Y/Slides/Infinite_Geometric_Sum,_Proof_Without_Words.pdf there is a "proof without words" of the sum of the infinite geometric series. However, I don't understand what makes the proof valid. In what order were the constructions done etc.? What...

Dear friends, I have divided the time into slots of fixed size. And i toss a coin of probability of heads 1/2 in the first slot. In the next slot, i toss a coin of probability of head 1/4, and in the i^th slot i toss a coin of prob of head 1/2^i. I do this until i get a head. What is the...

Homework Statement A ) A company produces microchips. It has some in storage and produces 34 an hour. After 1 hour it has a total of 3428 microchips i) How many chips will the company have a week later, assuming the production continues 24 hours a day? ii)An order is put in for 13,526...

Now for the proof of convergence/divergence of the geometric series we first deduce the Nth partial sum which is given by: \frac{r(1-r^n)}{1-r} Now for 0<r<1 this become \frac{1}{1-r} which clearly converges by AOL At r>1 it's similarly obvious why it diverges. But at r=1, I'm a bit...

I've spent all day on this problem and am wasting precious time needed for other work - please give any input you can! The question: given two wages, w1 and w2 where w2 > w1... a. the difference between the wages as a proportion of the lower: a = (w2 - w1) / w2 b. the difference between the...

Homework Statement A ball is dropped from a height of 24m and rebounds to a height of 16m. If each time it rebound two-thirds of the previous height, find the total distance traveled by the ball. Homework Equations The Attempt at a Solution I thought this might be a problem...

Find the sum to infinity of the series $1 +2z +3z^2+4z^3+...$

Hi Guys, Long time reader first time poster... This simple question has stumped me all day and I think I've finally cracked it! I'm hoping someone can confirm that, or tell me how wrong I am - either is fine :) One in 1000 cows have a rare genetic disease. The disease is not contagious...

Question: Two faulty machines, M1 and M2, are repeatedly run synchronously in parallel (i.e., both machines execute one run, then both execute a second run, and so on). On each run, M1 fails with probability p1 and M2 with probability p2, all failure events being independent. Let the random...

Question: Two faulty machines, M1 and M2, are repeatedly run synchronously in parallel (i.e., both machines execute one run, then both execute a second run, and so on). On each run, M1 fails with probability p1 and M2 with probability p2, all failure events being independent. Let the random...

Hello, Second term of a geometric series is 48 and the fourth term is 3... Show that one possible value for the common ratio, r, of the series is -1/4 and state the other value. ar=48, ar^3= 3... so ar^3/ar=3/48 which simplifies to r^2 = 1/16, therefore r = 1/4 Can anyone explain where...

Rather than arithmetic ("plus or minus") uncertainties, are there classical (not of Heisenberg uncertainty principle) measurements whose uncertainties otherwise appear as geometric ("times or divided by")?

Homework Statement Find a closed form for the nth partial sum, and determine whether the series converges by calculating the limit of the nth partial sum. 1. 2+2/5 + 2/25+...2/5k-1 Homework Equations The Attempt at a Solution What I did was I found out it was a geometric...

Homework Statement A large cube of glass has a metal reflector on one face and water on an adjoining face (the figure). A light beam strikes the reflector, as shown. You observe that as you gradually increase the angle of the light beam, if Theta is greater than 58.7 no light enters the water...

I just sent some time dicking around with the MacLaurin expansion of exp(-z2) to derive a series expression for √π, by integrating term-by-term along the real line. I'm not really concerned with wether this is a useful or well-studied expression, I just thought it would be a fun exercise...

Homework Statement A geometric series had first term 54 and 4th term 2. (i) What is the common ratio? (ii) Find the sum to infinity of the series. (iii) After how many terms is the sum of the series greater than 99% of the sum to infinity? Homework Equations N/A The Attempt at a...

Homework Statement Determine if the sequence {an} below converges or diverges. Find the limit of each convergent sequence an = n!/nn Hint: Compare with 1/n . Find the limit of the sequence {an} if it converges. I missed the lesson on factorials, and the book is useless. Sorry...

If our sun is the source of illumination, how can an object be observed from the Earth at full phase? Wouldn't the Earth eclipse the object? So then why can we see a full moon during full phase? Is it because the moon's orbit is inclined wrt to the Earth-Sun orbit? If so then wouldn't this by...

Homework Statement How do you find the median of the geometric distribution? Homework Equations M is median if P(X>=M) >= 1/2 and P(X<=M)>=1/2. The Attempt at a Solution I have found this inequality using the geometric series: (m-1)*log(1-p) >= 1/2

Homework Statement here is the series: \sum^{\infty}_{n=0}x(-15(x^{2}))^{n} Homework Equations The Attempt at a Solution I know that -1<-15x^{2}<1 for convergence (because of geometric series properties) but I run into a problem here: -1/15<x^{2}<1/15 You can't...

Generalizing past the quadropole moment-- geometric understanding of the octopole+ I'm having a bit of trouble articulating my question, but I hope the explanations will help you to understand the source of my confusion: The mono, di, and quadropole moments are all geometrically...

Homework Statement Posted this thread earlier but had mis read the given answer. please disregard older thread as I don't know how to delete it! Write down the condition for the numbers p, q, r to form an arithmetic sequence & geometric progression. Homework Equations \ a_n =...

Homework Statement Problem H-10. We will compute the mean of the geometric distribution. (Note: It's also possible to compute E(X^2) and then Var(X) = E(X^2)−(E(X))^2 by steps similar to these.) (a) Show that E(X) = (k=1 to infinity summation symbol) (k
*q^k−1*
p) where q = 1−p. (b)...