Sum Definition and 1000 Threads

[FONT=times new roman]Hi members of the forum,[FONT=times new roman] [FONT=times new roman]Problem: The real numbers $\displaystyle \alpha$, $\displaystyle \beta$[FONT=times new roman] satisfy the equations $\displaystyle \alpha^3-3\alpha^2+5\alpha-17=0,$ [FONT=times new roman] $...

Homework Statement Say we have an infinite sequence of natural numbers A such that no K subsequences can be found adjacent such that the average of the elements in any subsequence is equal for all K subsequences. Sorry about my poor description, an example would be that {2, 3, 4, 1} wouldn't...

Suppose you have many forces acting on an object, and the object moves in space in some time interval. Each force has done some work on the object. Suppose you took all these values for work, added them up, (they are all scalars). You'd obtain a scalar equal to the net work done on the object...

Express e^x from 1 to 8 as a Riemann Sum. Please, check my work? :) 1. Express ∫1 to 8 of e^xdx as a limit of a Riemann Sum. (Please ignore the __ behind the n's. The format is not kept without it...) _____n 2. lim Ʃ f(xi)(Δx)dx x→∞ i=1 Δx= (b-a)/n = 8-1/n = 7/n xi= 1 + 7i/n ____n lim Ʃ...

So I missed a class and am trying to figure out a question in my textbook but am completely lost. It goes a little something like this: Let f(x)=x3 and let P=<-2,0,1,3,4> be a partition of [-2,4]. a) Compute Riemann Sum S(f,P*) if the points <x1*,x2*,x3*,x4*>=<-1,1,2,4> are embedded in P...

Problem: Prove that for $n>0$, \sum_{i=0}^{n} (-1)^i \binom{n}{i}=0 Attempt: This seems clearly like a proof based on induction. 1) Base case: for $n=1$, \sum_{i=0}^{1}(-1)^i \binom{1}{i}=(-1)^0 \binom{1}{0}+(-1)^1 \binom{1}{1}=1-1=0 2) Show that $n=k$ being valid implies $n=k+1$ is valid...

\sum_{i=1}^{n} |(y_i-\theta)|=n\theta where theta is a fixed constant and y_i is a discrete random variable. does anyone know how to rewrite in close form?? also, everytime i use latex it starts a new line. how can i fix this so i can type directly with my sentances? thanks

Homework Statement cot^-1 3 + cot^-1 7 + cot^-1 13+... Homework Equations The Attempt at a Solution I first tried to write the nth term of the series t_n = cot^{-1}\left( 2^n + (2n-1) \right) Then I tried to calculate the limit as n→∞. But I simply can't do that. I mean I...

Dear Math Help Boards, I have a tricky problem that I hope one of you can help me with. (It's for a personal project, nothing to do with school.) I'm looking for a closed-form expression for the sum of the first through m-th terms of a combinatorial number. For those of you unfamiliar with...

Dear Physics Forums denizens, I have a tricky problem that I hope one of you can help me with. (It's for a personal project, nothing to do with school.) I'm looking for a closed-form expression for the sum of the first through m-th terms of a combinatorial number. For those of you...

Homework Statement I am trying to find the sum of the series in the attachment. Homework Equations The Attempt at a Solution I have tried to use various series and their derivatives, to not much avail. I am not sure how to handle the n^2 factor. Should I break it down to two...

Happy new year. All the best. I have one question. Is it true? \sum^{\infty}_{k=0}a_kx^k=\sum^n_{k=0}a_{n-k}x^{n-k} I saw in one book relation \sum^{\infty}_{k=0}\frac{(2k)!}{2^{2k}(k!)^2}(2xt-t^2)^k=\sum^{n}_{k=0}\frac{(2(n-k))!}{2^{2(n-k)}((n-k)!)^2}(2xt-t^2)^{n-k} Can you give me some...

Hi everyone! How to solve this: S(n) = { (a+b)/n } + { (2a+b)/n } + { (3a+b)/n } + ... + { (na+b)/n } where {x} represents fractional part of x. a,b,n are natural non-null numbers and (a,n)=1. I don`t need only an answer, i need a good solution. Thanks!

Homework Statement Let z1, ... zn be the set of n distinct solutions to the equation zn = a where a is a complex number. (a) By considering distinct solutions as the sides of a polygon in an Argand diagram show that these sum to zero.(b)...

Why in the attached picture is it legal to interchange the sum and integral? Is it just because n is not dependent on t? note: (c1)n is just a function of n

Hello, my question is that almost all textbooks say that a linear system will give the output to a weighted sum of impulses which equals the superposition of scaled responses to each of the shifted impulses. But if we apply the same input which is a weighted sum of impulses to a non linear time...

Homework Statement $$\sum_{i=1}^{m} \sum_{j=1}^{n}[2(x_i-x_{i-1})-3(y_j-y_{j-1})]$$ Homework Equations Multiple-sigma notation.The Attempt at a Solution I agree this seems like basic summation stuff, but i do not agree with the given answer. So, here is what I've done. $$\sum_{i=1}^{m}...

I choose a random number p_1 \in [0,1) and a subsequent series of (increasingly smaller) random numbers p_i \in [0, p_{i-1}). Then I can calculate the sum \sum_{i=1}^\infty p_i. Naturally, this sum is dependent on the random numbers chosen, so its particular result is not very insightful...

How do I evaluate the sum of this sigma notation problem? 20 ∑ k k=10 Normally, I would think to use the theorem for the sum of the first n integers: n ∑ k = n(n+1)/2 k=1 I don't know how to do this, however, since the lower limit is k=10, not k=1. My professor wrote this note on the board...

Hey, So I have a sum of complex numbers that really should be easy, but I'm not getting the right solution. It is with respct to using the Gram Schmidt process U1 = (i, -1, i) U2 = (1,1,0) So I perform the Gram Schmidt with U1 being my initial vector selection and I get: V2 =...

Homework Statement I am asked to prove that e^{iB} is unitary if B is a self-adjoint matrix. The Attempt at a Solution In order to prove this I am attempting to show e^{iB} \widetilde{e^{iB}} = 1. Using the assumption that B is self-adjoint I have been able to show that e^{iB}...

Homework Statement Is it possible to add the following subspaces: W_1 = Sp{(1,0,0)} and W_3 = Sp{(0,1,-1), (0,0,1)}? Homework Equations The Attempt at a Solution Will their sum be: Sp{(1,1,-1),(1,0,1)}?

Number 11) This is what I did: Ʃτ = F2r2 + F1r1 = 0 (195)(7) + F1(0.7) = 0 F1(0.7) = -(195)(7) F1 = -1365/0.7 F1 = -1950 N F1 = 1950 N Is that answer right? Number 12) This is what I did: Since this is a massless rod and the location of the axis is through the end, I = ML2 Linitial =...

Homework Statement Ʃ(A-i)/(N+1-i) sum of i=1 to N Homework Equations How do I solve this series for all 0<N<A cases. This series is the sum of N geometric variables of changing probability. I'd appreciate any help

Homework Statement Express the following as a definite integral: Express the attached limit as an integral. The Attempt at a Solution I have gotten as far as every part of the answer except the upper bound. the answer is: 10 [SIZE="5"]∫(from 1 to 10) [x-4lnx]dx 1 since the...

For what complex numbers, x, is Gn = fn-1(x) - 2fn(x) + fn+1(x) = 0 where the terms are consecutive Fibonacci polynomials? Here's what I know: 1) Each individual polynomial, fm, has roots x=2icos(kπ/m), k=1,...,m-1. 2) The problem can be rewritten recursively as Gn+2 = xGn+1 +...

Homework Statement I'm reading from the first edition of Axler's Linear algebra done right. In the section on sums of vector subspaces, he states: U = {(x,0,0) ∈ F3 | x ∈ F} W = {(y,y,0) ∈ F3 | y ∈ F} and 1.7 U + W = {(x,y,0) ∈ F3 | x,y ∈ F} However, shouldn't the answer be U...

$\sum\limits_{n = 1}^{\infty}\left(\sqrt{1 + n^2} - n\right)$ $$ \sqrt{1 + n^2} - n = \frac{1}{\sqrt{1 + n^2} + n} $$ Now what?

$\sum\limits_{n = 2}^{\infty}n^p\left(\frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}}\right)$ where p is any fixed real number. If this was just the telescoping series or the p-series, this wouldn't be a problem.

Calculate 5 \Sigma r=0 r(r+1) (Sorry I don't know how to do the proper notation online) How do you calculate the sum for this since the common difference is changing? I tried to write them out separately so the sum of r x sum of r+1 but I don't know how you put that in the sum formula. Sn =...

Homework Statement I have this series 1^{3}-2^{3}+3^{3}-4^{3}+5^{3}-6^{3} + \ldots Homework Equations and sequence of partial sums for this series that is: S_n = \sum_{k=0}^{n}(-1)^{k+1} k^3 = \dfrac{1 + (-1)^n(4n^3 + 6n^2-1)}8 =\begin{cases} \dfrac{2n^3+3n^2}4; & n \text{ is...

Homework Statement Let V be a real inner product space, and let v1, v2, ... , vk be a set of orthonormal vectors. Prove Ʃ (from j=1 to k)|<x,vj><y,vj>| ≤ ||x|| ||y|| When is there equality? Homework Equations The Attempt at a Solution I've tried using the two inequalities given to us in...

Homework Statement Find the sum of the sequence: 2, -2/3, 2/9, -2/27, 2/81, . . . Homework Equations The Attempt at a Solution I can see that the number is multiplied by -1/3, but I'm unsure of how to find the sum. Any pointers?

Why oh WHY do the arrows in this TORQUE problem keep alternating between SINE and COS This is what they did: I understand the whole Torque = F * r, but WHY (at the top of the diagram) is the arrow pointing up = (25 N)cos 30 and NOT sin and why is the arrow pointing to the right...

Lagrange's four-square theorem states that any natural number can be expressed as the sum of four integer squares. I've noticed that the first few values of 8n-1 can all only be expressed as a minimum of the sum of four squares. Is this true for all values of n? What's the proof behind it? Thanks.

Hi, I have determined, correctly I believe, that the following series converges: 1/[(3n-2)(3n+1)] Now I am asked to determine its sum. I have tried separating it into two subseries, but each time got a p-series with p=1, hence to no avail. The answer should be 1/3, but how may it be...

I teach cost-benefit analysis, which requires me to teach monte carlo simulation for sensitivity analysis. I use excel. I understand how to generate a number with uniform, triangular, normal or other distributions, but I don't know how to randomly generate a set of numbers between zero and one...

Homework Statement Homework Equations I really wish they existed in my notes! *cry*. All I can think of is that integrating or in other words summing the dirac delta functions for all t, would be infinite? None the less the laplace transform exist since its asked for in the question and i...

Find the sum of \Sigma 200 r=5 5r-2 Sn = n/2 [2a + (n-1)d ]I used S 200 and I got about 101400 but then when I verified on my calculator it was 100058, my calculator has the sigma notation for working out the sum of , how do you get 100058?

i'm trying to prove the sum of nth roots of unity = 0, but I don't really know how to proceed: suppose z^n = 1 where z ε ℂ, suppose the roots of unity for z are 1, ω, ω^2, ω^3 ... ω^n the sum of these would be S = 1 + ω, ω^w, ω^3 +...+ ω^(n-1) + ω^n from here I had an idea to do some...

Homework Statement Estimate the area of the region under the curve y = ln(x) for 1 ≤ x ≤ 5. Use the left-hand rule with n = 50. Homework Equations The Attempt at a Solution Do I really have to do 50 calculations? There has to be a faster way :/ (aside from using the definite...

Homework Statement Show that \sum_{k=0}^{\infty} \sqrt[k]k-1 converges. Homework Equations Ratio, radix theorems, comparison with other sums... The Attempt at a Solution No idea whatsoever. Where does one begin in this case ? With other cases I'm quite confident.

Does there exist a sequence of real nonzero numbers whose sum converges to 0? I would think there isn't, but I'm interested in people's opinions and arguments. For any nonzero m, a series of nonzero numbers whose sum converges to m can easily be constructed using the formula: \sum...

My answer is almost correct, except for the negative sign. Can anyone help? Many thanks. Homework Statement Q. Without using a calculator, show that \sin10^{\circ}+\sin80^{\circ}=\sqrt{2}\cos35^o Homework Equations The Attempt at a Solution...

Homework Statement I am trying to prove how \(g''(r)=\sum\limits_{k=2}^\infty ak(k-1)r^{k-2}=0+0+2a+6ar+\cdots=\dfrac{2a}{(1-r)^3}=2a(1-r)^{-3}\). I don't know what I am doing wrong and am at my wits end. The Attempt at a Solution (The index of the summation is always k=2 to infinity)...

Homework Statement Let X,W,Y be iid with a common geometric density f_x(x)= p(1-p)^x for x nonnegative integer and p is in the interval (0,1) What is the characteristic function of A= X-2W+3Y ? Determine the family of the conditional distribution of X given X+W? Homework Equations...

Homework Statement let y_1 and y_2 be iid bin(5,1/4) random variables let v=y_1+2*y_2 and u = 3*y_1 -2y_2 find f_uv (u,v) and the cov(u,v) Homework Equations f_y (y) = (5 choose y) (1/4)^y (3/4)^5-x for x=0,1,2,3,4,5 covariance=E(uv)-E(u)E(v) The Attempt at a Solution...

$$ 2\frac{1}{3} + \frac{1}{3^2} + 2\frac{1}{3^3} + \frac{1}{3^4} + 2\frac{1}{3^5} + \cdots = \frac{1}{3}\sum_{n = 0}^{\infty}\left(\frac{1}{3}\right)^n $$ I am stuck on what to add into account for the 2 at every other term.

Homework Statement Please write a specific function to define this series. Also provide a sum that the series converges to.Homework Equations Sn - {1, 1+1/e2, 1+1/e2+1/e4, 1+1/e2+1/e4+1/e6, ...} The Attempt at a Solution I know that the common ratio is 1/e2 and that you can raise that to...

f(x,y) = (1/x) for 0≤y≤x≤1 A new rv Z=X+Y where X,Y not independent find the pdf of z My approach F(z) = P(Z≤z) = ∫∫fXY(x,y) dx dy x= -∞ to ∞ y= 0 to z-y f(z) = d/dz(F(z)) = ∫fXY(z-y,y) dy y= -∞ to ∞ (using Leibnitz) where i am stuck is this doesn't converge