Sums Definition and 349 Threads

I'm working on a calculus project and I can't seem to work through this next part... I need to substitute equation (2) into equation (1): (1): r\frac{\partial}{\partial r}(r\frac{\partial T}{\partial r})+\frac{\partial ^{2}T}{\partial\Theta^{2}}=0 (2): \frac{T-T_{0}}{T_{0}}=A_{0}+\sum from n=1...

Homework Statement Normed space (l^\infty,\|\cdot\|_\infty) with subspace S\subset l^\infty consisting of convergent sequences x=(x_n)_{n\in\mathbb{N}}. Given a sequence of maps A_n:l^\infty\to\mathbb{R} defined as $$A_n(x)=\sup_{i\in\mathbb{N}}\frac{1}{n}\sum_{j=0}^{n-1}x_{i+j}$$need to...

Homework Statement Use the Fourier series technique to show that the following series sums to : 1+\frac{1}{3^2}+\frac{1}{5^2}+...=\frac{\pi^2}{8} Homework Equations The Attempt at a Solution Don't know what the first few steps are...but I assume that I need to first express the sum as...

Hi. I'm currently tutoring this student with High school math, and I'm completely stumped on this question that he was asked on his test. I'm hoping the community can help me help my student! Homework Statement The student was presented with two sums of a geometric sequence (eg, Sum of...

Hi there, This is going to be difficult for me to explain, so I will try my best. My statistics is kind of rusty... I've been given census data to analyze and I'm having problems. The totals are all given by the government. For a specific attribute I've been given rows of different areas...

Hi everyone. In a proof I'm working on, I have a ratio of two sums of functions in the following form: \frac{f_1(x)+f_2(x)+...f_n(x)}{g_1(x)+g_2(x)+...+g_n(x)} I want to prove this ratio is monotonically increasing in x. All of the functions f_i(x) and g_i(x) are positive and also...

Homework Statement If R = [0,4]x[-1,2], use a Riemann sum with m=2, n=3 to estimate the value of ∫∫(1-xy^2)dA. Take the sample points to be the lower right corners. Homework Equations NoneThe Attempt at a Solution 2*1[f(2,-1) + f(2,0) + f(2,1) + f(4,-1) + f(4,0) + f(4,1)] = some value Just...

Is anyone familiar enough with excel to use it to find the sum in of sigma notation when "n" is large? \Sigma^{40}_{k=1} \sqrt\frac{3k-3}{4} Something like this?

Would someone be kind enough to explain Jacobi sums in a simple manner using actual numbers. I have read over the math jingo 100 times and have no clue what it actually does. Thanks! Edit: Here is a link to the wiki of the Jacobi sums. http://en.wikipedia.org/wiki/Jacobi_sum

Homework Statement Let \mathbb{R}*=\mathbb{R}\{0} with multiplication operation. Show that \mathbb{R}*=\mathbb{I}2 ⊕ \mathbb{R}, where the group operation in \mathbb{R} is addition.Homework Equations Let {A1,...,An}\subseteqA such that for all a\inA there exists a unique sequence {ak} such that...

Use a geometric or algebraic argument to find a formula for the partial sums $A_n$ of an arithmetic sequence. I know that the partial sum is $S_n = n/2(2a_1+(n-1)d)$ where d is the difference. $A_n = \sum\limits_{k = 1}^n a_k$ I can come up with $n/2(a_1+a_n)$ but how do I get the difference?

I have seen double sums, but have come across a problem involving sums over primes. However, this sum is inside a second sum, and is taken over all primes that divide the outside index, like this: \sum_{k=1}^{n} \sum_{p | k} \frac 1p for p prime. Is there any way to manipulate this...

In 'Linear Algebra Done Right' by Sheldon Axler, a direct sum is defined the following way, We say that V is the direct sum of subspaces U_1, \dotsc ,U_m written V = U_1 \oplus \dotsc \oplus U_m, if each element of V can be written uniquely as a sum u_1 + \dotsc + u_m, where each u_j \in U_j...

Homework Statement \int_0^2 x^2 \, dx using true definition involving Riemann Sums (w/o fundamental theorem). Homework Equations I don't know what the relevant equations may be, perhaps some type of lim\sum f(x)(x_{j}-x_{j-1}) The Attempt at a Solution No attempt. Just seeking the...

I have some questions and doubts in trigonometry. I hope somebody can solve these questions. Q1) If for real values of x, cos\theta = x +\frac{1}{x}, then a) \theta is acute angle b) \theta is right angle c) \theta is an obtuse angle d) no value of \theta is possible I will post the following...

Proposition: \sum_{i=0}^{p-1} (\frac{i^2+a}{p})=-1 for any odd prime p and any integer a. (I am referring to the Legendre Symbol). I was reading a paper where they claimed it was true for the a=1 case and referred to a source that I don't have immediate access to. So I was wondering if...

I am currently reading about riemann sums and several different sources uses these abbreviations, inf and sup, and I am not certain what they mean. Could someone explain them to me?

I am currently reading about finding areas under graphs using summations, specifically taking the of the number of rectangles, n, goes to infinity. My books says that "because the same limit value is attained for both minimum value f(mi) and the maximum value f(Mi), it follows from the squeeze...

. . Let \ \ p_n \ \ = \ \ the \ \ nth \ \ prime \ \ number.Examples:p_1 \ = \ 2 p_2 \ = \ 3 p_3 \ = \ 5 p_4 \ = \ 7- - - - - - - - - - - - - - - - - - - - - - - - - - - - Let \ \ n \ \ belong \ \ to \ \ the \ \ set \ \ of \ \ positive \ \ integers. Prove (or disprove) the following:p_n \ +...

So I was just working through Courant's calculus and am a bit confused as to where a few variables are pulled out of. Homework Statement Integration of f(x) = x We can see that a trapezoid is formed, so the relevant equation: 1/2(b-a)(b+a) is the value of this integral. To confirm that our...

My lecturer has said that beppo levi means for and increasing sequence of Xi where Xi is simple for all i, it holds that ∫limi → ∞XidP = limi → ∞∫XidP But why is it that he later says things like ∫ limi→ ∞ Ʃin=1P2(Bw1n)dP1(w1) = limi → ∞Ʃin=1∫P2(Bw1n)dP1(w1) is a result of beppo...

1. Homework Statement Write an m.file that will integrate a function f(x, y) on any given rectangle (a,b)\times(c,d) and returns the value of the integral from a to b and c to d of the function f(x,y) . Include error-catching code in the case that the integral diverges. The program...

So kind of like this thread, I'm looking to convert a discrete sum to an integral. My idea thus far has been to arrive at a function via spline interpolation. I'm doing a few different types of sums, but the first ones look like \displaystyle a=\sum_{i=1}^{100}{data[1]*data[4]} where data...

Homework Statement I've seen two methods that prove the integral test for convergence, but I fear they contradict each other. Each method uses an improper integral where the function f(x) is positive, decreasing, and continuous and f(x) = an. What confuses me is one method starts off the...

Homework Statement So, I know the pdf for independent random variables is found by using the convolution; the pdf is f[sub:X+Y](a) = ∫ f[sub:X](a-y)f[sub:Y](y)dy, but can I just use the density function for a function of a random variable instead; that is: f[sub:X+Y](x[u,v], y[u,v])(Jacobian...

Dear forumers, I have a question about taking direct sums and products of state spaces in QM. Picture I have a state space that describes two (indistinguishable) particles which is a direct sum of two one-particles spaces: \epsilon_t = \epsilon_1 \oplus \epsilon_2 Furthermore, picture that...

I've noticed lots of interesting properties of the patterns of numbers in the Fibbonacci sequence that can be expressed as the sum of two squares. In fact, it's what got me into number theory in the first place. There seem to be no two adjacent entries that are not the sum of two squares- and it...

1 \ - \ \frac{1}{2} \ + \ \frac{1}{3} \ - \ \frac{1}{4} \ + \ ... \ - \ \frac{1}{n - 1} \ + \ \frac{1}{n} \ - \ \frac{1}{2n + 1} \ < \ ln(n), where n is a positive odd integer I worked this out (rediscovered it) and proved it by induction. For example, when n = 71...

Homework Statement Show that the generating function A(x) = \sum_n a_n x^n of a_n = \sum_{k=0}^n {n+k \choose 2k} 2^{n-k} satisfies A(x) = \frac{1-2x}{4x^2-5x+1}Homework Equations The Attempt at a Solution A hint was given to "interchange the sums". After doing that, I don't see how to...

I would like to find a nice formula for \sum_{k=0}^{n - 1}ar^{4k}. I know that \sum_{k=0}^{n - 1}ar^{k} = a\frac{1 - r^n}{1 - r} and was wondering if there was some sort of analogue.

Homework Statement I have the series 1^3+2^3+3^3...n^3, and need to find a formula containing n to represent the sum of the nth terms. The motivation is to find a conjecture, which I can then prove using mathematical induction. The Attempt at a Solution I see that n=1 , 1^3=1...

Evaluate using telescoping sums: (a) $\sum_1^\infty\frac{(-1)^{n-1}}{n(n+2)}$ (b) $\sum_1^\infty\frac{1}{n(n+k)}$, $k$ integer $>0$ My attempt: (a)$\frac{1}{n(n+2)}=\frac{1}{2}\left(\frac{1}{n}-\frac{1}{n+2}\right)$ Adding the terms for $n$ even, we get...

I have: $Z=X_1+\ldots+X_N$, where: $X_i\sim_{iid}\,\text{Exponential}(\lambda)$ $N\sim\,\text{Geometric}_1(p)$ For all $i,\,N$ and $X_i$ are independent. I need to find the probability distribution of $Z$: $G_N(t)=\frac{(1-p)t}{1-pt}$ $M_X(t)=\frac{\lambda}{\lambda-t}$...

Homework Statement Z=X_1+\ldots+X_N, where: X_i\sim_{iid}\,\text{Exponential}(\lambda) N\sim\,\text{Geometric}_1(p) For all i,\,N and X_i are independent. Find the probability distribution of Z Homework Equations G_N(t)=\frac{(1-p)t}{1-pt} M_X(t)=\frac{\lambda}{\lambda-t}...

i am trying to re-express the following in terms of a rational function: \frac{(0+x+2x^2+3x^3+...)}{1+x+x^2+x^3+...} . i know that this is supposed to be \frac{1}{x-1} but I can't figure out how to do it. I know the denominator is just \frac{1}{1-x}. so in order for this work out, the...

Hi, I am reading a paper, and at some point the authors claim that: \sum_{m=1}^{L+1}\frac{\prod_{\substack{l=1\\l\neq m}}^{L+1}\frac{\lambda(m)}{\lambda(m)-\lambda(l)}}{\lambda^r(m)}=0 the question is HOW? Any tiny hint will be highly appreciated. Thanks

Never Mind I answered my own question two minutes after posting it. I don't know how to take this question down so I just deleted it.

This is a very similar question to what I posted earlier. Basically I am trying to find when (x+y)6 = x6 + y6 assuming that xy≠0 I am trying to play with it algebraically to find a contradiction, but have been unsuccessful I'm also working on (x+y)7 = x7 + y7 assuming xy≠0 I'm...

Hi, I keep seeing indirect uses of a result which I think would be stated as follows: If a module M over the unital associative algebra A is written M\cong S_1\oplus\cdots\oplus S_r (where the S_i are simple modules), then in any comosition series of M, the composition factors are, up to...

Homework Statement Use the form of the definition of the integral to evaluate the following: lim (n \rightarrow ∞) \sum^{n}_{i=1} x_{i}\cdotln(x_{i}^{2} + 1)Δx on the interval [2, 6] Homework Equations x_{i} = 2 + \frac{4}{n}i Δx = \frac{4}{n} Ʃ^{n}_{i=1}i^{2} =...

Is it possible to have an solution to this sort of integral? And if not, why not? \int_0^\infty \frac{e^{-ax}}{e^{-bx}+e^{-cx}}dx Is a Taylor expansion the only way forward? Many thanks David

Is it possible to find a non-recursive formula for the partial sums of a convergent series?

I have read somewhere that we can extend the notion of a series of a sequence \sum_{i=1}^{\infty} a_n to sums over an arbitrary index set, say a : I \to \mathbb{R} is a family of real number indexed by I, then \sum_{i \in I} a_i is the sum of all the elements. I think the text...

I'm reviewing my Calc 1 material for better understanding. So, I was reading about the area under a curve and approximating it using Riemann sums. Now, I understand the method, but I was a little confused by finding xi*. I know there is a formula for it xi*=a+Δx(i). What does the "i" stand for...

I have the following homework to do. Apologies if it seems very easy - I just had a knee surgery and I think I can't really think straight due to pain medication, I feel so fuzzy and sleepy and my damn knee still hurts like $#%&. So instead of representing numbers in a binary way I need to...

Hey, I'm trying to prove a larger theorem; in order to complete my proof I need to use the following lemma (or, if it turns out to be false, try a completely different method of proof): Consider any two sets of n nonzero integers, A and B. If their respective sums and products are equal...

Homework Statement Let \sum_{n=1} a_n and \sum_{n=1} b_n be convergent series. For each n \in \mathbb{N}, let c_{2n-1} = a_n and c_{2n} = b_n. Prove that \sum_{n=1} c_n converges. Homework Equations The Attempt at a Solution Not sure whether the following solution is...

Homework Statement Establish the identity: 1+cos(2θ)+cos(4θ)+cos(6θ)=4cosθcos(2θ)cos(3θ) Homework Equations cos(a)+cos(b)=2cos((a+b)/2)cos((a-b)/2) The Attempt at a Solution I understand how to do a simple cos(+/-)cos problem according to the Sums as Products equations, but I am...

Homework Statement https://www.physicsforums.com/attachment.php?attachmentid=39642&stc=1&d=1317853920 how do you go about solving this? Homework Equations i have proved the binomial theorem.The Attempt at a Solution i was considering cases, for j(even or odd). would this be the right direction?

Can someone give me an example of a bounded function f defined on a closed interval [a,b] such that f does not attain its sup (or inf) on this interval? Obviously, f cannot be continuous, but for whatever reason (stupidity? lack of imagination?) I can't think of an example of a discontinuous...