Fraction Definition and 653 Threads

Hello! (Wave) I want to show that if $c_k=\frac{p_k}{q_k}$ the $k$-th convergents of the continued fraction $[a_0; a_1, a_2, \dots, a_n]$, then $q_k \geq 2^{\frac{k-1}{2}} (1 \leq k \leq n)$. Could you give me a hint how we could show this? (Thinking)

I'm given the problem 14x+5y=31, 2x-3y=-29 This is a set of linear equations that I am asked to solve. That being said I am getting hung up on how to distribute a fraction. The answer is x=9, however, I am getting x=36 and the step I do not understand is distributing the fraction. We then get...

Hi, I would like to expand the following expression: 1/[((a+s)*(1+b/s)^m)], where a, b, and s are real nonnegative values and m is an arbitrary positive integer. Particularly, according to partial fraction expansion, it becomes: Sum[A_j/[(1+b/s)^j],{j,1,m}]+B/(a+s). I look for a closed-form...

Homework Statement Consider solar neutrinos of energy 1 MeV (EDIT: 10 MeV not 1 MeV) which are formed at the center of the sun in the ##\nu_2## eigenstate. What fraction of it do you expect to arrive at Earth as ##\nu_\mu## and what fraction as ##\nu_\tau##? Assume that it evolves adiabaticaly...

Homework Statement Find Partial Fraction Expansion 10/[s (s+2)(s+3)^2] Homework EquationsThe Attempt at a Solution 10/[s (s+2)(s+3)^2] = A/s + B/(s+2) + C/(s+3)^2 + D/(s+3) A = 10/[(s+2)(s+3)^2], s approaches 0 = 10/(2*3^2) = 5/9 B = 10/[s (s+3)^2], s approaches -2 = 10/(-2) = -5 C =...

When Cern is done with a particle beam, what fraction of a typical beam bunch is left having not scattered out of the bunch? Is it of order 1/2? A very rough estimate would do. Edit, I should add that I read an accelerated beam might be used for a day and at the end of the day they dispose of...

Homework Statement Find the partial fraction decomposition for: ##\frac{1}{\left(x^2-1\right)^2}## Homework EquationsThe Attempt at a Solution Please see my attached images. I think the image shows my thought process better and it would take me well over an hour to type all that out! Im...

There is a problem in a PreCalculus book that I'm going over that states: Express the sum ##\frac{1}{2⋅3}+\frac{1}{3⋅4}+\frac{1}{4⋅5}+...+\frac{1}{2019⋅2020}## as a fraction of whole numbers in lowest terms. It goes on to state that each term in the sum is of the form...

I’ve had huge difficulty understanding/interpreting the concept of branching fraction. So correct me if I’m wrong please: Let’s take the decay of Ra-226 with half-life of 1602 years as an example. It decays through alpha 1 chanel to the excited state of Rn-222 with E=0,187 Mev ( branching...

Homework Statement If mole fraction, pressure fraction and volume fraction are denoted by Xmol , Xp, XV respectively, of a gaseous component, then what is the relation between them? Homework Equations mole fraction = mole of component / total moles pressure fraction = pressure of component /...

Conceptually, at least, this is a simple question, although I recognize that it might be hard to calculate in practice from available data. The matter-energy budget of the universe is measured (in a model dependent way) to consist of a certain percentage of dark energy, a certain percentage of...

Hi i’m having problems with the following equations: X(w)=2/(-1+iw)(-2+iw)(-3+iw) This then becomes the following equation according the the tutorial, although there is no explanation as to how: X(w)=1/-1+iw, -2/-2+iw, +1/-3+iw The commas indicated the end of each fraction to make it easier...

Assume a source of EM radiation at wave length λ hits a barrier with a small circular hole of diameter d << λ. What fraction of the radiated power (watts) that hits the hole passes through it? Does it depend on the thickness of the barrier? I understand that after passing through the hole, the...

I have a problem to put the complex number in mode (1000/3, ∠36.87), apparently the division simbol gives some syntax error

Someone once told me that a fraction with a big denominator is a small fraction but no mathematical reason was given. So, 1/45 < 1/2. Why?

Homework Statement Homework EquationsThe Attempt at a Solution Honestly speaking , I have absolutely no idea about this problem .This doesn't happen often .In the KTG chapter , only definition and formula of Average , RMS and Most probable speeds is given . This question is for an...

- check if right check if right Now, 2 seems to be the right answer for A yet when i made x=5 and subtracted new form form the old one I got a difference of ~$\frac{4}{9}$ (should be 0 obviously) I got A=2 B=$\frac{45}{21}$ C=2 How to calculate $\lim_{{x}\to{\infty}}(- e^{-x})$

Homework Statement \int\frac{x^2}{\sqrt{x^2+4}}dx Homework Equations n/a The Attempt at a Solution Letting x=2tan\theta and dx=2sec^2\theta d\theta \int\frac{x^2}{\sqrt{x^2+4}}dx=\int\frac{4tan^2\theta}{\sqrt{4+4tan^2\theta}}2sec^2\theta d\theta=\int\frac{8tan^2\theta...

Homework Statement When completely hollow. To what fraction of volume should water be poured into it so that it is just enough to cause the sphere to sink. [/B]Homework Equations Archemedes : Buoyant fource equals weight of water displaced The Attempt at a Solution Let's first find the...

Hi Point kinetics equation with one-group delayed neutrons, dn/dt= ((ρ-β)/∧)n+λC dC/dt= (β/Λ)n - λC When a reactor in steady-state operation, derivative terms would be zero. From second equation, C/n= β/λΛ is found. For given β=0.0065, λ=0.08 and Λ=0.0001, the ratio of the steady state...

Let me start by saying that this is from a 30 question assessment on Big Oh, Big Theta, and Big Omega. I understood every other question, however, even after being given the correct answer, I do not understand why my answer was wrong for this one. If you could point me in the direction of any...

Homework Statement Hi, So I had a pretty long question solving a Linear ODE but now I've gotten stuck at this stage where I can't seem to get it into the right form to carry out partial fraction expansion Homework EquationsThe Attempt at a Solution [/B] I'm quite sure that I what I have at...

Homework Statement [/B] When you drink cold water, your body must expend metabolic energy in order to maintain normal body temperature ( 37.0°C ) by warming up the water in your stomach. Could drinking ice water, then, substitute for exercise as a way to “burn calories?” Suppose you expend 430...

Homework Statement This question is problem 1.3.1 from Herbert Callen's textbook in Thermodynamics, second edition. In the answers I have the following: Mole fraction of NaCl = 0.057, molar volume =$18\times 10^{-6} m^3/mole$. How did they arrive at these answers? did they multiplied the...

Hey guys, I am seriously confused by this problem. 3/4 [ 5/6 ( -18/25 ) + 1/2 ] I would appreciate it if someone shows me the step by step process. Thanks in advance :)

If $x_1,x_2,...,x_{10} \in [0;\frac{\pi}{2}]$ and $\sum_{i=1}^{10}\sin^2x_i = 1.$ - then prove, that: \[ \frac{\sum_{i=1}^{10}\cos x_i}{\sum_{j=1}^{10}\sin x_j} \ge 3\]

Homework Statement The width of a rectangle is increased by one tenth, but the area remains the same. By what fraction has the length of the rectangle been reduced? The Attempt at a Solution Length = x Width = x + 1/10 Set equation: L * W x *(10x + x) 10x^2 + x^2 I am stuck... Please help...

Homework Statement I would like to understand whether reocurring formulas in the example solutions to my homework problems hold in general and ideally where they're derived from. (I'm new here; if there's there a way to embed formulas in f.i. latex code please tell me.) As an example problem...

I am finding it hard to understand this questions its sounds more like fraction subtraction than multiple but the army test sample questions says its multipication You have 3 1/4 boxes of paper. You give 1/2 to the paper to a colleague, how many boxes of paper do you have left. The test says 1...

Homework Statement If a proton with a kinetic energy of 6 MeV is traveling in a particle accelerator in a circular orbit of radius 0.75m, what fraction of its energy does it radiate per second? m = 1.67 * 10^-27 epsilon_0 = 8.854 * 10^-12 c = 3 * 10^8 Homework Equations dE/dt = (q^2 a^2) /...

Homework Statement See below Homework EquationsThe Attempt at a Solution I am looking at a particular integral, and to get started, my text gives the indication that one should use partial fraction decomposition with ##\displaystyle \frac{\cos (ax)}{b^2 - x^2}##. Specifically, it says "then...

I have known the interaction parameter of a certain liquid mixture which has the phase behaviour can be described by the lattice model. , is it possible for us to know the temperature at the critical point? Besides, if we know the temperature at the critical point, can the volume fraction of...

Homework Statement Find the partial fraction decomposition of ##\displaystyle \frac{1}{x^4 + 2x^2 \cosh (2 \alpha) + 1}## Homework EquationsThe Attempt at a Solution Using the identity ##\displaystyle \cosh (2 \alpha) = \frac{e^{2 \alpha} + e^{- 2\alpha}}{2}##, we can get the fraction to the...

Hello Everyone , I need some help in solving this partial fraction $\frac{x^2}{(x-2)(x+3)(x-1)}$ I am using this method in which the partial fraction is broken into 3 parts namely A,B &C $\frac{x^2}{(x-2)(x+3)(x-1)}=\frac{A}{(x-2)}+\frac{B}{(x+3)}+\frac{C}{(x-1)}$...

I have figured out a nice way to prove that if the complex numbers z_1,z_2,\ldots, z_N\in\mathbb{C} are all distinct, then the equation \prod_{n=1}^N \frac{1}{z - z_n} = \sum_{n=1}^N \frac{\alpha_n}{z-z_n} is true for all z\in\mathbb{C}\setminus\{z_1,z_2,\ldots, z_N\}, where the alpha...

$\tiny{206.08.05.44}$ $\textsf{Use the method of partial fraction decomposition}\\$ \begin{align*} \displaystyle I_{44} &=\int \frac{4x^3+6x^2+128x}{x^5+32x^3+256x}dx\\ &=4\int \frac{1}{x^2+16} \, dx +6\int \frac{x}{(x^2+16)^2} \, dx +64\int \frac{1}{(x^2+16)^2} \, dx \end{align*}$\textsf{so far...

$\tiny{206.8.5,42}\\$ $\textsf{partial fraction decomostion}\\$ \begin{align} \displaystyle && I_{42}&=\int\frac{3x^2+x-18}{x^3+9x}\, dx& &(1)&\\ && \frac{3x^2+x-18}{x^3+9x} &=\frac{Ax+B}{x^2+9} +\frac{C}{x} & &(2)& \end{align} $\textit{just seeing if this is set up ok before finding values} $

Hi guys, I have a question that I can't seem to wrap my head around. This is the question: A power reactor is fueled with slightly enriched uranium. At the end of core life (i.e., when it is about to be batch refueled), 30% of the power comes from the fissioning of the built up Pu-239...

Hi, I have the following question: 2 whole number ¼ - 1 whole number 2/3 ------(1) 1 whole number (1/4 -2/3) I am getting –ve sign which is wrong. However if I do: 2 whole number ¼ + 1 whole number 2/3 Then: 3whole number (1/4 * 3/3 + 2/3 * 4/4) = 3 whole number(3/12 + 8/12) = 3...

How to convert mass fraction to mole fraction? Is it using the molecular weight?

Take the fractions 2/5 and 4/10. 2/5 cannot be simplified further without converting the integers into decimals. I need to know the name of the 2/5 fraction. Maybe it's called irreducible form or something like that. Let me give another example, of 1/3, 3/9, or 33/99 only 1/3 is irreducible...

$\sqrt{5} * \left(\frac{5}{4}\right)^{\!{\frac{1}{2}}}$ Many Thanks :)

I want to create a binary matrix (${X}_{m*n}$) containing $C$ ones ($||X|| = C$). Additionally I want to have the number of elements of each row ($m$) to have an exponential form. This is: for each row the number elements needs to be equal to $e^{alpha * i}$ or in symbols: ||${X}_{i}$|| =...

Homework Statement Consider a gas of molecules of mass m at equilibrium at temperature T. Obtain an expression for the fraction of molecules having kinetic energy e = 1/2mC2 in the range e to e + de. This is problem 5.3, page 48 of Vincenti and Kruger's Intro to Physical Gas Dynamics Homework...

$\frac{2a}{a^2-4} - \frac{1}{a-2}$ Now using difference of squares in $a^2-4$ = $\left(a+2\right) \left(a-2\right)$ Then the fraction would be $\frac{2a}{\left(a+2\right) \left(a-2\right)} - \frac{1}{a-2}$ Can this fraction be simplified further? Many Thanks :)

This problem is a little elementary, If we were to multiply the fraction $\frac{3}{7}$ by two which way should I be using, $2\left(\frac{3}{7}\right)=\frac{6}{14}$ --------------- 1 or $\frac{2}{2}\left(\frac{3}{7}\right)=\frac{6}{14}$ --------------- 2 I usually multiply fractions using...

Hey, all! I'm learning partial fraction decomposition from Serge Lang's "A First Course in Calculus." In it, he gives the following example: \int\frac{x+1}{(x-1)^2(x-2)}dx He then decomposes this into the following sum: \frac{x+1}{(x-1)^2(x-2)} =...

Homework Statement I did some homework and wolfram alpha afterwards gave different result than what I have. Apparently I must have done some error, but I couldn't find error after I doublechecked about two times. So I decided to see if you guys can find an error. And what is the correct answer...

Homework Statement Proof that: ##\lim_{x \to 1} \frac{1}{1+x} = \frac{1}{2}## using the epsilon-delta definition of a limit. (Problem below) Homework Equations ##\lim_{x \to 1} \frac{1}{1+x} = \frac{1}{2} \iff \forall ε>0, \existsδ>0, \forall x ∈ \mathbb{R}\backslash\{-1\}: 0 < |x-1| < δ...

Hi, \frac{3 }{ 2(x+1)} + \frac{1}{x+1} = \frac{5}{6} Can you help me on this problem. I'm not sure on how to solve this and please be kind enough to explain your steps little bitMany Thanks:)