Summation Definition and 627 Threads

Okay I've seen how crazy Riemann sums can get in real analysis and I've noticed a heirarchy of notation. The Stewart/Thomas etc... kinds of books use; \lim_{x \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x Where; \Delta x = \frac{b - a}{n} and x_i = a + i\Delta x Then the books like Apostol and...

I tried integration then applying limit as n tends to infinity, for k = 1, it becomes a circle, but as k increases, points decrease hence it should be wrong.

Hi I got to the following equation while going through a book. I can't figure out how the second line comes from the first. Can anyone please help me understand? (1/2*\sum_{q=-Q}^Q V_s,q .H(w_q) .exp(iw_q t))^n is written as, 1/2^n * \sum_{q1=-Q}^Q \sum_{q2=-Q}^Q ... \sum_{qn=-Q}^Q...

Hello, If we are given a gaussian function which is continuous in x we know that: \int_{-\infty}^{+\infty}e^{-x^2}dx=\sqrt{\pi} What if the gaussian function is discrete in x? What is the result of \sum_{x=-\infty}^{+\infty}e^{-x^2} = \\? where x\in \mathbb{Z}

Homework Statement By first expressing the general term in partial fractions, find an expression in terms of n for summation of r=2 to n ( 1 / (r^2 - 1) ). Hence show that summation of r=1 to n ( 1 / r^2) i less than 7/4 for all values of n Homework Equations The Attempt at a...

Hello, I'm having some issues solving some apparently 'basic' summation problems where they give you a couple summations and you derive the missing summation. I would appreciate any help not only solving this particular question but actually understanding the situation. Thanks...

Homework Statement evaluate \sum\frac{1}{e^n} from 0 -> infinity Homework Equations N/A The Attempt at a Solution from what I've learn, i can calculate summation i in form \sumna ,a is integer or \sum f(n+1)-f(n) but how to make 1/e^n in any those form? can give me any clue please...

Calculate the summation of i=1 to inf of the summation of j=i to inf of p^(j+i). Yes, it is the summation of a summation. p^(j+i) can be separated into (p^j)*(p^i).

Homework Statement Ok so I'm meant to answer: To what scalar or vector quantities do the following expressions in suffix notation correspond? (expand and sum where possible). 1) aibjci 2) aibjcjdi 3) dijaiaj 4) dijdij 5) eijkaibk 6)eijkdij Homework Equations The...

So the question asks: What is the value of the "summation of" 2n+1/3n from "n=1 to infinity." I changed 2n+1/3n into 2*(2/3)n so i could use it as a geometric series. So now i just use the rule "a/(1-r) = sum" where a = first term and r = ratio i get 2/(1-(2/3)) which = 6. The answer is...

Hello, Could you help me derive this function, so I can find the minimum of it. z=\sum_{i=1}^{n}{\sqrt{\left( x-x_{i} \right)^{2}+\left( y-y_{i} \right)^{2}}} Thank you.

Homework Statement Hi guys, i have a exercise of the limit of a summation that is the formal definition of definite integral and i need resolve and explain, but i can't resolve for the rational exponent, for this, need help, thanks in advance. Homework Equations \lim_{n \rightarrow...

\sum^k_{n=1}e^{-n\sum^k_{n=2}...e^{-n\sum^k_{n=k-1}e^{-n}}} Can anyone help me find out if this converges and if so how to calculate the sum? I don't have an idea on how to even start. This is not homework

Hello, I think I am fundamentally confused with summation convention. For example, if I have \epsilon_{ijk}x_j\delta_{jk} Can I sift the levi civita and get \epsilon_{ijj}x_j=0 or sift x and get \epsilon_{ijk}x_k\not=0 Each gives a different answer. What mistake...

I am working on a problem that uses the notation: \sum_{i,j=1}^n A_{i,j} Where A is an (n x n) matrix. I am a little unsure of what the summation is over, due to the odd notation "i,j = 1". My first guess is that this is shorthand for \sum_{i=1}^n \sum_{j=1}^n A_{i,j} But I...

So I am trying to derive a formula from one of the standard summation formulas except starting at a different index. So if I have the series.. \sum i = \frac{n(n+1)}{2} Where "i" runs from 1 to n. (I don't know how to put it in the code.) If I want to make the series start from zero, I...

hi i am just reading some notes on tesor analysis and in the notes itself while representing vectors in terms of basis using einstein summation notation the author switches between subsripts and superscripts at times. are there any different in these notation. if so what are they and when should...

how do I evaluate \sum_{k=0}^d \binom{n+d-k}{n} ?

what is the sum of this series ?? \sum_{n=0}^{\infty}n^{b}e^{ian} for every a and b to be Real numbers from the definition of POlylogarithm i would say \sum_{n=0}^{\infty}n^{b}e^{ian}= Li_{-b}(e^{ia}) however i would like to know if the sum is Cesaro summable and what it would be...

Homework Statement \sum^{n}_{r=1}(\lg \frac{2^r(r+1)}{r})=\sum^{n}_{r=1}[\lg 2^r+\lg (r+1)-\lg r] Homework Equations The Attempt at a Solution i found the answer to be (2^n-1)\lg 2 +\lg (n+1) Am i correct , or it can be further simplified ? Thanks .

Homework Statement A set contains numbers from 1-100. What is the least non-negative value that one can form by putting a + or - in front of each number, and summing the values?Homework Equations there are a few general summation formulas which I know... The Attempt at a Solution The...

Homework Statement Basically need to use einstein's summation convention to find the grad of (mod r)^n and a.r where a is a vector and r = (x,y,z) Homework Equations The Attempt at a Solution Not sure where to begin really.. :S grad (mod r)^n= (d/dx, d/dy, d/dz) of root (X1^2...

Hi, I've been looking through my algorithms book/notes and I've come across this summation I'm not quite sure how they got to. \sum^{lgn - 1}_{i = 0}\frac{n}{lgn - i} = n\sum^{lgn}_{i = 1}\frac{n}{i} where lgn = log_{2}n, it's just to make it simpler any clue? cheers,

Homework Statement It can be shown that ∑(n=1) to (n=∞) of 1/n² = π²/6 use this fact to show that ∑(n=1) to (n=∞) of 1/(2n-1)² = π²/8 Homework Equations The Attempt at a Solution

Hello, I am having difficulty approaching this problem: Assume that K, Z_1, Z_2, ... are independent. Let K be geometrically distributed with parameter success = p, failure = q. P(K = k) = q^(k-1) * p , k >= 1 Let Z_1, Z_2, ... be iid exponentially distributed random variables with...

\sum_{i=1}^{n} i is the sum of all numbers between 1 and n. I'm trying to find one for odd numbers where you need to find the sum of all odd numbers between 1 and n. I tried 2n+1 which worked, only for first n numbers, not for numbers 1 to n. Thanks for the help.

Hello all, In the book "A First Course in General Relativity" by Schutz (1985 Edition) in chapter 2 there is a problem concerning summation that has me confused. Note: This is not homework, just an interest of mine. The given quantities are: A = (5,0,-1,-6) B = (0,-2,4,0) C = [ 1 0 2 3...

Homework Statement This isn't really a homework question, but something I've been wanting to know out of curiosity in David Griffiths' Introduction to Electrodynamics. On pages 131 and 132, there is a Fourier series, V(x,y) = \frac{4V_0}{\pi}\sum_{n=1,3,5...}\frac{1}{n}e^{\frac{-n \pi...

Hello this is something that just crossed my mind: For every real sequence (a_n)_{n\geq 1} we can define the generating function A(z)=\sum_{n=1}^\infty{a_nz^n}. and this definition suggests that we can compute the sum of the sequence by evaluating A at 1: A(1)=\sum_{n=1}^\infty{a_n}...

If I have ∑(k xi + j yi)...how will I apply the distributive law on it?...I mean how do you split this notion?

Homework Statement find the sum for \sum_{k=1}^{\infty} kx^{k} Homework Equations \sum_{k=0}^{\infty} x^{k} = \frac{1}{1-x}; -1 < x < 1 The Attempt at a Solution \sum_{k=1}^{\infty} kx^{k} = \sum_{n=0}^{\infty}(n+1)x^{n+1} = x\sum_{n=0}^{\infty} (n+1)x^{n} = x \frac{d}{dx}...

\sum_{k=0}^{\infty}a_{k}(k+r)(k+r-1)x^{k-1} I need to get my x term to look like xn. If I set n = k-1, then that makes my index start at n = -1, which is silly. What can I do?

Homework Statement Begin with an equilateral triangle T of side length 1 At the middle of each side of T place an equilateral triangle whose side lengths are 1/3 Repeat this process ad infinitum By summing an appropriate series, show that the area A of the fractal obtained above is finite...

Homework Statement Define Tn as the sum of the first n terms, for various values of a and x, e.g. T9(2,5) is the sume of the first nine terms when a = 2 and x = 5. The first n terms are 0-10, including both 0 and 10. Homework Equations T0=1, T1= (xlna)1/1, T2= (xlna)2/2!, T3=...

Homework Statement Ok I have the answer to a question, all the working is given, however, I'm having trouble following it. Homework Equations http://img695.imageshack.us/img695/426/answer.jpg The Attempt at a Solution I am completely lost, could someone please explain the steps that have...

Homework Statement Consider a neuron with resting potential of -65 mV and threshold of -55 mV. It receives two synaptic inputs with similar synaptic conductances, one with reversal potential of -10 mV and the other with reversal potential of -58 mV. Draw the predicted postsynaptic...

Homework Statement Let lim n-> ∞ a_n = L. Then, let f(x) = ∑ from n=0 to ∞ of (a_n)(x^n). Show that the lim x-> 1 (1-x)f(x) = L. Homework Equations The Attempt at a Solution This one is pretty far over my head. I know at some point you're supposed to use Abel/SBP, but here is what I have so...

Please look at the attachments (for the problem and my work) , thank you. The answer from the book is 2.84 but I got 14.4. How come? My problem is #23

I hope can someone clarify this for me. I have a sequence f(of n) which is like this: fn(x) = 0-- if--x<\frac{1}{n+1} is = sin^2(x/pi)--if--\frac{1}{n+1}<=x<=\frac{ 1}{n} is = 0--if--\frac{1}{n}<x (the - are for spaces because I don't know how to do it. Nothing is negative) Then...

Homework Statement I'm just not sure how to change the operators in summation, can anyone help? Let s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k what is s_{2n}?Homework Equations s_{n}=\sum_{k=1} ^n ((-1)^{k+1})/k The Attempt at a Solution s_{2n}=\sum_{k=1} ^{2n} ((-1)^{2k+1})/2k or...

Homework Statement what is the summation of a function where n=1 to n=infinity? For example, given a function sin[(pi)nt]. Homework Equations The Attempt at a Solution I asking how I get that I do not know what should I do

Just got a "thought experiment" question from a colleague. The question, as phrased was: If an audio signal was composed by adding all of the frequencies in the audible range, what would it sound like? I thought it was interesting, so I attempted to solve it by integral. My calculus skills...

How to say ... Hi ... I'm doing a small presentation and I was wondering how I would say the following summation: \sum_{0<i_1<...<i_n<p} \left(\frac{i_1}{3}\right) \frac{(-1)^{i_1}}{i_1 i_2 \cdot \cdot \cdot i_n} where \left(\frac{i_1}{3}\right) is the Legendre symbol, n is a positive odd...

Hey guys and gals, While this technically isn't homework, I figured this is the place to post. I am working over a problem and I am at a point in the solution that has me a bit stumped. Perhaps someone may provide some guidance. In acoustics, we run into the problem of a radiating body...

I just thought I would share this, I was about to ask you fine people how to do this when I realized the square root of the sum of progressive to regressive data equals the highest point. I.E. 1+2+3+4+5+6+5+4+3+2+1=36 6^{2}=36 and I tried this a few times and the results were the same...

Homework Statement n+1 , n \sum (nCk-1) f(x)g(x) + \sum (nCk) f(x)g(x) k=1 , k = 0 (for first and second summations respectively) I can't just say that that is equal to: n+1 \sum (n+1 C k) f(x)g(x) k=0...

Homework Statement I want to calculate a sum (where the end value is in the sum), eg: \sum^{n-1}_{i=1}{2i+n} Homework Equations I don't want to 'split' the sum, i just want to write this. The Attempt at a Solution syms i n for n=1:5 for i=1:n-1...

I came across this approximation in a book. I am not sure why this approximation is valid.. \frac{1}{N}\sum_{n=0}^{N-1}n.sin[4\pi f_o n + 2\phi] \approx 0 f_o is not near 0 or 1/2 Saurav

Homework Statement Prove by induction the following summation formula: \frac{1}{1\1*2} + \frac{1}{2*3} + ... + \frac{1}{n(n+1)} = 1 - \frac{1}{n+1} n \geq 1 Homework Equations - The Attempt at a Solution Inductive step: 1. \frac{1}{1*2} + \frac{1}{2*3} + ... + \frac{1}{n(n+1)} +...

Hi everyone. I hardly remember the fomulas of summation of sequence. I got this problem. {\frac{1}{8}}\sum^{\infty}_{n=2}n({\frac{3}{4}})^{n-2} The result is 2.5. How can I solve this problem? Thanks all. :)