Sum Definition and 1000 Threads

Prove that if $\displaystyle \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=1$, then $\displaystyle \frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}=0$

Hi all, Say that I already know W1, W2 are both subspaces of a vector space V, W1∩W2={0}, and that dim(W1)+dim(W2)=dim(V)=n, can I thus conclude that V=W1+W2, namely V is the direct sum of W1 and W2?

Homework Statement As part of a problem I have to show that lim_{n\to\infty}\sum_{i=\frac{n}{2}}^{n}\frac{1}{i}=ln(2) Homework Equations Taylor expansion of ln(2): \sum_{i=1}^{\infty}\frac{(-1)^{k+1}}{k} The Attempt at a Solution ln(2) can be written as: ln(2) =...

Homework Statement Ʃ(x-->infinity, x>0) 11/(n(n+2)) Homework Equations The Attempt at a Solution I am not quite sure what to do, i think that i am supposed to put it into partial fractions. i changed it to the form 11/(n^2+2n) --> 11/((n+1)(n+1)-1)) --> 11/((n+1)^2-1)...

(1/1-1/6) + (1/2-1/7) + (1/3-1/8)...+(1/95-1/100) This was on the cover of a local paper with the caption "You can't solve this but he can!" The 'he' in the caption was a 12 year old boy. On the next page they gave as his solution 2.2. Two things are clear, 1.-the boy had the right answer and...

My question arises in the context of bosonic string theory … calculating the number of dimensions, consistent with Lorentz invariance, one finds a factor that is an infinite sum of mode numbers, i.e. positive integers … but it really goes back to Euler, and his argument that the sum of all...

I am curious if there is a universal formula to find all possible sums of a given number. For instance, to add to 10: 1+9 2+8 1+1+8 2+2+2+3+1, etc I came up with a simple algorithm, but I'm sure there is something similar to Gauss's formula which can be utilized. I have heard Partitions...

Hi all, I was just looking at the U.S. electoral map, and I was wondering if there could possibly be a tie in presidential elections (the answer is probably no). I tried to think of an efficient algorithm to answer this question, but due to my limited intelligence and imagination, all I...

Hi all, Can anyone gimme any clues to solve the sum below (or solve it outright :D)? \sum_{i=k}^{n} \frac{i!}{(i-k)!} I'm trying to solve one of Feyman's logic problems (bored geek alert) and I'm stuck at this point. And since my high school days are so far behind... Thanks in...

Homework Statement Find a value for n for which the nth partial sum is ensured to approximate the sum of the alternating harmonic infinite series to three decimal places. Homework Equations Sn = Ʃ(-1)^k+1*1/k = 1 - 1/2 + 1/3 - 1/4 + 1/5 - . . . S1 = 1 S2 = 1 - 1/2 S3 = 1 - 1/2 + 1/3 S4...

Homework Statement I just have to prove the well known identities: \cos(\alpha + \beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta) \sin(\alpha + \beta)=\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin( \beta) But the thing is that I've to use the Taylor power series for the sine and cosine...

Hello everybody, I'm new to this forum so thanks for having me. I'm trying to find the times when the extrema occur for a periodic wave f(t) equal to the sum of three sine waves. Given f(t) = sin(2∏at) + sin(2∏bt) +sin(2∏ct) where a, b and c are whole numbers in lowest form (i.e...

I always see problems like "how many structurally distinct abelian groups of order (some large number) are there? I understand how we apply the theorem which tells us that every finite abelian group of order n is isomorphic to the direct sum of cyclic groups. We find this by looking at the...

Homework Statement Let a_{n} be an alternating series whose terms are decreasing in magnitude. How to compute the sum as precisely as possible using four-digit chopping arithmetic? In particular, apply the method to compute \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{(2n)!}}} and...

How to prove the convergence or divergence of ? $$\sum^{\infty}_{n=1}\frac{\Gamma(n+\frac{1}{2})}{n\Gamma{(n+\frac{1}{4})}}$$

Problem: Let A, B, C be three non-collinear points. Let D, E, F be points on the respective interiors of segments BC, AC and AB. Let θ, φ and ψ be the measures of the respective angles ∠BFC, ∠CDA and ∠AEB. Prove IAS(ABC) < θ +φ + ψ < 540 - IAS(ABC).(IAS means internal angle sum). Now I am...

Hi members of the forum, Problem: Rationalize the denominator of $\displaystyle \frac{1}{a^\frac{1}{3}+b^{\frac{1}{3}}+c^{\frac{1}{3}}}.$ I know that if we are asked to rationalize, say, something like $\displaystyle \frac{1}{1+2^{\frac{1}{3}}}$, what we could do is the following...

Suppose we have two multivariate functions, u_{1}(x,t) and u_{2}(x,t). These functions are solutions to second-order linear equations, which can be written as follows: Au_{xx}+Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+Fu=G Each of the coefficients are of the form A(x,y). Now, the linearity of these...

I'd really appreciate some help with a sum of: a_n= |sin n| / n All I've thought of, is that I should probably create a subsequence of {a_n}, such that all the elements of this subsequence {a_n_k} are >epsilon >0, and then compare the subsequence to 1/n which diverges. However, I have no...

Homework Statement Find the sum of 5^1-5^2+5^3-5^4+...-5^{98} a. (5/4)(1-5^99) b. (1/6)(1-5^99) c. (6/5)(1+5^98) d. (1-5^100) e. (5/6)(1-5^98) Homework Equations The Attempt at a Solution I feel as though this is actually a simple problem and that I'm not looking at it the right way. [5^1...

the answer is 3^2048. How do I get there?

Homework Statement The Attempt at a Solution Alright, so, I'm clueless about doing this one. I do know that it's extremely similar to e^x \sum_{n->0}^{\infty}\frac{x^n}{n!} But really, that means nothing! Usually there's another function/series I can compare and then integrate, in...

If I have a periodic function that is say a sum of a number of sine functions I can use a Fourier Transform to get the component functions. Now, if I have a bunch of square waves of differing amplitudes and frequencies that I add up into a resultant waveform. Given that waveform what'd be...

Hi members of the forum, Problem: The real numbers $\displaystyle \alpha$, $\displaystyle \beta$ satisfy the equations $\displaystyle \alpha^3-3\alpha^2+5\alpha-17=0,$ $ \displaystyle \beta^3-3\beta^2+5\beta+11=0.$ Find $\displaystyle \alpha + \beta$. This problem has me stumped. It's...

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 ∫(from 1 to 10) [x-4lnx]dx 1 since the definition of...

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...