Sums Definition and 349 Threads

Homework Statement At my old university, Calculus was taught much differently than it is where I am now. My old school focused on numerical things, which this school focuses much more on pictures, abstract, etc. and it's very difficult for me. At my old school, we were given a shape...

Homework Statement I'm working through an example from class and the textbook, but I'm confused about how the steps progress mathematically. The example involves the Gibb's partition for a paramagnet. \sum_{s} exp(\beta \mu B \sum_{i}^{N} s) Where s = -a,-a+1...a for each spin...

Hello. I have to solve some integrals using both the standard theorem of calculus and infinite Riemann sums. \int_{1}^{7} (x^2-4x+2) dx = \lim_{n \to \infty } \sum f(x_i)\Delta x_i = \lim_{n \to \infty } \sum (x_i^2 - 4x_i + 2)6/n Evaluating the definite integral results in an answer of 30...

It is well known that the Harmonic Series diverges (1/1+1/2+1/3+1/4+...), but that the series [1/1+1/4+1/9+1/16+...] converges. In the first series, the denominators are the integers, whereas in the second example, the denominators are the integers to the power of 2. My question is, at what...

1) if you have 2 line segments making a right angle, and connect the endpoints with a line segment with an outward curve relative to the vertex, will the area inside always be irrational? 2) if you have 2 line segments making a right angle, and connect the endpoints with a line segment with an...

Hello, i just came accros: Sum(i) , from i=1 to i=n which apparently equals n(n+1)/2 -Is there a way to derive this from the sum, or you just have to use your intuition and think through what exactly is being summed and the range of summation? -Do you have any resources to offer, that...

Homework Statement Learning about sums of subspaces and wanted to be sure that I am understanding this correctly. Say that I have two subspaces of R^2: U = {(x,y) in R^2 : y + 2x = 0} W = {(x,y) in R^2 : y - 3x = 0} and I wanted to geometrically (and algebraically) represent their...

I am looking at a geometric series problem that has already been worked out, so not assigned, but I do not see where they get a number: Summation from n=1 to inf: 1/(n^2+4n+3) In doing the partial sums, he has (1/2)* summation... 1/(i+1) - 1/(i+3) I understand the breakup, but where does...

Can anyone help me with the below question? for each of the following pairs of random variables X,Y, indicate a. whether X and Y are dependent or independent b. whether X and Y are positively correlated, negatively correlate or uncorrelated i. X and Y are uniformly distributed on the disk...

Homework Statement Suppose that X is uniformly distributed on (0,2), Y is uniformly distributed on (0,3), and X and Y are independent. Determine the distribution functions for the following random variables: a)X-Y b)XY c)X/Y The Attempt at a Solution ok so we know the density fx=1/2...

show that if all the row sums of a matrix A belong to C (nxm) are zeroes, then A is singular. Hint. Observe that Ax=0 for x=[1 1 ...1]T

Homework Statement Use a left and midpoint Rienman sum to approximate \int_{0}^{3} x^3 dx The Attempt at a Solution Left \Delta x = \frac{3}{n} Now here is the problem, for left points, what is my 'a'? I can't carve up my intervals and this isn't a right-end approximation so I...

Homework Statement http://img842.imageshack.us/img842/1404/unledro.png Now what I don't understand is this part [PLAIN][PLAIN]http://img821.imageshack.us/img821/7479/unledoq.png Is he trying to say that \frac{1}{2n^3} = R_n? Is that how he deduced that \frac{1}{2n^3} < 0.0005?

Homework Statement Identify an=the summation from k=1 to n of (2n)/(4k2+1) as a Riemann sum of an appropriate function on an appropriate interval and find the limit as n approaches infinity of an. Homework Equations There is no interval givien so I assume its from 0 to 1. The...

Homework Statement Let f_n(z)=\sum_{k=0}^n\frac{1}{k!}z^n. Show that for sufficiently large n the polynomial f_n(z) has no roots in D_0(100), i.e. the disk of radius 100 centered at 0. Homework Equations This is a sequence of analytic functions which converges uniformly to e^z on C...

Homework Statement Translate the following equation into canonical S of P form: F = (xyz' + x'w)(yz + x'z') Homework Equations DeMorgan's theorem? (x + y)' = x'y' (xy)' = x' + y' The Attempt at a Solution Converting from Sum of Products to Products of Sums requires the following...

Homework Statement Convert the Riemann's Sum to an integral: (1/50) * [(sqrt(1/50)) + (sqrt(2/50)) + (sqrt(3/50)) ... + (sqrt(50/50))] Homework Equations The Attempt at a Solution (1/50) times Integral (upper limit 1 and lower limit 0) of sqrt(x) dx Is my solution correct?

HELP!Sums of Random Variables problem: Statistics Homework Statement 3. Assume that Y = 3 X1+5 X2+4 X3+6 X4 and X1, X2, X3 and X4 are random variables that represent the dice rolls of a 6 sided, 8 sided, 10 sided and 12 sided dice, respectively. a. If all four dice rolls yield a 3, what...

Hello all, Is there a closed form expression for the convergence of C(s) = \sum_{n=1}^{N} n^{s} Cheers, Adrian

http://arxiv.org/abs/1102.1688 State Sums and Geometry Frank Hellmann PhD Thesis, 106 pages (Submitted on 8 Feb 2011) "In this thesis I review the definition of topological quantum field theories through state sums on triangulated manifolds. I describe the construction of state sum invariants of...

1. What causes the algebraic sum of the currents (signs included) flowing into each of the four junctions NOT equal to zero? 2. What causes the algebraic sum of the EMFs in each closed loop NOT equal to the algebraic sum of all the IR drops in each loop?For the first question, I thought that...

Homework Statement f: [0,1] -> Reals, f(x) = 3-x2 P={0,1/2,1} Find lower and upper Riemann sums, and approximate the definate integral using them and find the corresponding approximation error. Homework Equations The Attempt at a Solution So I tried finding the upper Riemann...

1. The problem statement, all variables and givennown data 1)FInd the nth left endpoint approximation Ln for f(x) = 3x^2-2 on [0,2]. What is the limit as n approaches infinity Ln in this case? 2)Evaluate: \sum45i=5 (2i-5) Homework Equations Ln = \sumNj=1 f(cj)(xj-xj-1) The...

How do I calculate the lattice sums A12 and A6 for a BCC structure? I have calculated the following so far: A12 = 8(1/1)^12 + 6(1/root2)^12 + 12(1/2)^12 + 16(1/root5.5)^12 + 8(1/root6)^12 = 8.097. Have i made any mistakes? Are my nearest neighbour values correct? Please help! o:)

I'm supposed to show that the sum C+C ={x+y,x,y in C}=[0,2] a) Show there exist x1,y1 in C1 for which x1+y1=s. Show in general for any arbitrary n in the naturals, we can always find xn, yn in Cn for which xn+yn=s. b) Keeping in mind that the sequences xn and yn do not necessarily converge...

Homework Statement 2/(n+8)(n+6); find the sum of the series with n=1 to infinity. Homework Equations Hmm.. The Attempt at a Solution I tried to split this into a partial fraction and try to find the limit, but for some reason that didn't work for me. I'm confused as to approach...

Homework Statement Sum from 0 to infinity of (2^n + 6^n)/(2^n6^n) Homework Equations No idea. The Attempt at a Solution I am completely dumbstruck on how to do this one. Could someone give me a hint on where to start? Thanks a lot!

Homework Statement Convert the following expression into sum of products and products of sums (AB+C)(B+C'D) Homework Equations Distributive Property The Attempt at a Solution for product of sums it would be (AB+C)(B+C'D) since it is already in this form. When calculating sum of...

I want to prove the following proposition: Given any uncountable set of real numbers S, there exists a countable sub-collection of numbers in S, whose sum is infinite. Please point me in the right direction.

Homework Statement Given F = (A'+B')[ABD' + A'C + A'BD], simplify into a simplest product of sums.Homework Equations The Attempt at a Solution Multiplying through gives me A'BD + A'C. I have tried expanding, adding consensus terms, but I'm not sure how to take it from there. Additionally, I...

Homework Statement I'm having some confusion about combining sums. Our goal when combining these sums is to have the, (x-c)^{\text{whatever}} term to be the same in both sums. My confusion is better explained in an example. (see below) Homework Equations The Attempt at a...

Homework Statement Use the midpoint rule to estimate the area under the graph of f(x) = 7/x and above the graph of f(x) = 0 from [1,25] using two rectangles of equal width. Homework Equations N/A The Attempt at a Solution So first I found \Deltax by using (b-a) / n and got (25 - 1)...

Homework Statement The question is to use upper and lowers sums, Un and Ln, on a regular petition of the intervals to find the integral from 1 to 2 of f(x) = [[x]], where [[x]] is the greatest integer function.Homework Equations \Deltax = \frac{b-a}{n} The Attempt at a Solution \Deltax =...

Let f be a Riemann integrable function defined on an interval [a,b], and let P = \{a = x_0 < x_1 < \ldots < x_n = b\} be a partition of [a,b]. I don't understand why the definition of (say) the upper Riemann sum of f associated with P is always given as U(f,P) = \sum_{i=1}^n M_i (x_i -...

Hi all, I have a question of computing the expectation of random sums. E(sim_{k=1}^N X_k) = E(N)E(X) if N and X_1, X_2,...are independent and X_k's are iid. Here both N and X_k's are r.vs. But the condition of N and X_1, X_2,...being independent is not true in many cases. How will...

Homework Statement Given the joint density, f(x,y), derive the probability density function for Z = X + Y and V = Y - X. Homework Equations f(x,y) = 2 for 0 < x < y < 1 f(x,y) = 0 otherwise. The Attempt at a Solution For Z = X + Y, I can derive the fact that, f_Z(z) =...

Homework Statement I have these three series here \sum_{n=2}^{\infty} a_n \cdot z^n - \sum_{n=1}^{\infty} a_n \cdot z^{n+1} + \sum_{n=0}^{\infty} a_n \cdot z^{n+2} what I would like to do is to add the together as one sum representation The Attempt at a Solution From what I...

Homework Statement Two parallel wires carry a current I and 2I in different directions. What is the magnetic field halfway between the two wires? Homework Equations Ampere's law Int (B dot dA) = permissivity x enclosed current The Attempt at a Solution Draw a circle around wire 1...

Homework Statement [PLAIN]http://img696.imageshack.us/img696/3438/46981606.jpg Homework Equations The Attempt at a Solution in those squars, I am sure about everything that i did and i get it wrong.. the only thing i don't know is bn+1 how would i know if its =, < or > than...

HI guys, I don't know where should I post this but well here it is! Can anyone tell me where to find some practice problems on physics, chem and math online? (I hope they are free and interesting!) I am in grade ( class) 10 in school. And since these are my vacations I thought of doing...

using an exponential regulator exp(-\epsilon n) the sum 1+2+3+4+5+6+7+...= -1/12+ 1/\epsilon^{2} and for Casimir effect 1+8+27+64+125+...= -1/120+ 1/\epsilon^{4} can i simply remove in the calculations of divergent series 1+2+3+4+5.. and similar the epsilon terms imposing...

Homework Statement If f:(-a,a)-->Real numbers, then f can be rewritten as the sums of an even and an odd function Let k: Real numbers\{-1}-->Real numbers be given by k(x)=\frac{x^2+4}{x+1} (i) Prove that there is no interval (-a,a) on which k is either even or odd (ii) Find an even...

Homework Statement If b>a, find the infimum of b - a for arbritrary b,a. Homework Equations - The Attempt at a Solution So this is for a darboux sum but I'm stuck on proving that 0 is the inf. It seems really easy but I can't seem to think straight. First thing is 0 is a lower...

I've been trying to tackle a problem of the following form lim_{n \rightarrow \infty} \sum_{k=0}^n f(k,x) I know that the limit of each function is zero as n goes to infinity. ie. lim_{n \rightarrow \infty} f(n,x) =0 But I'm not sure how to approach the problem above. I would greatly...

I'm stuck on how to approach the following problem. lim_{x \rightarrow \infty} \sum_{j=0} ^x e^{-j/x} Does anyone have any ideas?

Given linear transformations S: Rn --> Rm and T: Rn --> Rm, show the following: a) S+T is a linear transformation b) cS is a linear transformation I know that since both S and T are linear transformations on their own, they satisfy the properties for being a linear transformation, which is that...

Homework Statement FIND THE LOWER SUM FOR THE REGION BOUNDED BY f(x) = 25 - x^2 AND THE X - AXIS BETWEEN x = 0 and x = 5. SOLVE ANALYTICALLY! Homework Equations None that I'm aware of... The Attempt at a Solution f(x) = -x2 + 25 and \DeltaX = b-a/n = 5-0/n = 5/n Here's...

Homework Statement Prove that the integral from 0 to x of (sin t)/(t + 1) dt > 0. (Sorry I don't know how to use the integral latex.)Homework Equations We have only learned about lower and upper sums, and how the integral is equal to the supremum of lower sums and the infimum of upper sums when...

Homework Statement This is somewhat a repost... except I have figured out some of it and I have cleaned up the question. Your task is to estimate how far an object traveled during the time interval 0<= t >= 8 , but you only have the following data about the velocity of the object...

Homework Statement I really need help starting this problem as I am not sure what to do. Your task is to estimate how far an object traveled during the time interval 0t8 , but you only have the following data about the velocity of the object. time (sec) 0 1 2 3 4 5 6 7 8...