Summation Definition and 610 Threads

Prove that S_{n} = \frac{n}{2}(2a + L) where L = a +(n-1)d Can someone guide me through the proof please Thx

Einstein Summation Convention / Lorentz "Boost" Homework Statement I'm struggling to understand the Einstein Summation Convention - it's the first time I've used it. Would someone be able to explain it in the following context? Lorentz transformations and rotations can be expressed in...

I have the following recurrence that I am trying to come up with atleast a simplified version if not a closed form. T(n) = T(n-1) + \sum_{i=1}^{(n-1)/2} [(n-(i+1)) * (i-1) * 2 + 2] in addition if n is even I must add the following to T(n) ((n/2) - 1)^2If any of you can help that would be...

Hi, I'm having some trouble understanding what is meant when the limit of a summation is i<j. Does it mean the limits are i = 0 to i = j? Thanks!

Sum the following: Sin(x) + Sin(x+d) + Sin(x+2d)...+Sin(x+(n-1)d). I only know that summation of Sin and Cos functions whose arguments are in Arithmetic Progression can be done through telescopic series. But I don't know how to proceed. Please Help!

Homework Statement \frac{1}{n}\lim_{n\rightarrow\infty}\sum_{k=1}^{n}f(a+\frac{b-a}{n}k) The Attempt at a Solution I tried to solve it simply. \frac{1}{n}\lim_{n\rightarrow\infty}\sum_{k=1}^{n}f(a+\frac{b-a}{n}k)=\int_{0}^{1}f(a+(b-a)x)dx =f(b)-f(a)

Can someone check this solution. Homework Statement \lim_{n\rightarrow\infty}\sum^{n}_{i=1}\sqrt{\frac{1}{n^2}+\frac{2i}{n^3}} The Attempt at a Solution =\lim_{n\rightarrow\infty}\frac{1}{n}\sum^{n}_{i=1}\sqrt{1+\frac{2i}{n}}=\int^{1}_{0}\sqrt{1+2x}dx for u=1+2x->du=2dx...

Hi there, I'm trying to right a program for class that 1st assigns random single precission floats from 0 to 1 to the elements 1-d array and then sums them up. Next I'm supposed to compare to this thing called the Kahan summation algorithm for different values of N (array size) using the...

Hello all! In solving some math problems, I encountered the following sum: \sum_{k=1}^{r+1} kb \frac{r!}{(r-k+1)!} \frac{(b+r-k)!}{(b+r)!}. \quad \mbox{(eqn.1)} Now, I have asked Maple to calculate the above sum for me, and the answer takes a very simple form: \frac{b+r+1}{b+1}. \quad...

I'm interested in the problem: \sum_{n=1}^{ \infty} \frac{1}{n^3} and would like to know more about what attempts have been made at it and any insights into it but I am unable to find much because I don't know the name of this series or if it even has one. I have learned what little...

So... I want to find the Cauchy sum of the Taylor polynomial of \exp x \sin x. I know how to do this with maple, which only requires the command taylor(sin(x)*exp(x), x = 0, n). I can also try the good old f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+\cdots...

Is there such a thing? The factorial is usually defined as n! = \prod_{k=1}^n k if k is a natural number greater than or equal to 1. Is there an operation that is defined as \sum_{k=0}^n k if one wants to find, for instance, something like 5+4+3+2+1? I ask because I was thinking about...

After studying Cesaro and Borel summation i think that sum \sum_{p} p^{k} extended over all primes is summable Cesaro C(n,k+1+\epsilon) and the series \sum_{n=0}^{\infty} M(n) and \sum_{n=0}^{\infty} \Psi (n)-n are Cesaro-summable C(n,3/2+\epsilon) for any positive epsilon...

If an infinite discrete sum is calculated via integrating over a density of states factor, is this integral an approximation to the discrete sum? i.e the discrete sums could be partition functions or Debye solids.

Prove the following statement: \[ \sum\limits_{r + s = t} {\left( { - 1} \right)^r \left( \begin{array}{c} n + r - 1 \\ r \\ \end{array} \right)} \left( \begin{array}{c} m \\ s \\ \end{array} \right) = \left( \begin{array}{c} m - n \\ t \\ \end{array} \right) \] Any initial...

I don't see how the following works: \sum_{n=0}^\infty \delta ( n - n_0 ) z^{-n} = z^{-n_0} I am missing the steps from \sum_{n=0}^\infty \delta ( n - n_0 ) z^{-n} to z^{-n_0} . If I try this step by step: \sum_{n=0}^\infty \delta ( n - n_0 ) z^{-n} = \sum_{n=0}^\infty \delta ( n - n_0...

Hello, Can anyone give some hints on how to solve this: \sum_{n=0}^{K-1}\frac{sin(2\pi n^2\Delta)}{n} It's just the n^2 that complicates things. I tried re-writing it as Im\sum_{n=0}^{K-1}\frac{e^{j n^2 x}}{n}, where x=2\pi \Delta but I cannot solve this either. Thanks, svensl

I have an equation; f(x) = \sum_{i=1}^{x-1} s^i Where s is a constant. Is it possible to transform f(x) into continuous functions ? If so, how ?

Is \sum_{u,v} H_{i-u,j-v}F_{u,v} the same as \sum_u\sum_v H_{i-u,j-v}F_{u,v} [SIZE="6"]? (Don't worry about what H,F,i,j,u,v are. I'm only asking about the notation.)

(This is not a homework question!) I have no education in this kind of math yet, but I wonder how many ways you are allowed to use the summation sign sigma. I can't seem to get a good explanation on google or wikipedia. Since I like to try myself with tex, I will write an example of it...

Find \sum_{1}^{n} \tan(a f_{n} ) \cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - \cdots \sin\left( x \right) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \tan(x) = \sin(x) / \cos(x) There might be equations for the summation of a series of sine functions or an equation...

I was wondering if anyone could tell me more about the Riemann Zeta function, esp at negative values. Especially when \sum_{n=1}^{\infty}n= \frac{-1}{12} R where R is the Ramanujan Summation Operator. Could anyone post a proof?

I was reading through a proof of the summation formula for a sequence of consecutive squares (12 22 + 32 + ... + n2), and the beginning of the proof states that we should take the formula: (k+1)3 = k3 + 3k2 + 3k + 1 And take "k = 1,2,3,...,n-1, n" to get n formulas which can then be...

I have this HW problem: Suppose Un and Vn are sequences of positve numbers such that the ratio of Un+1/Un will always we less than Vn+1/Vn. Show that 1) If Vn converges Un converged and 2) If Un diverges, Vn diverges. I did the first part by showing that for any n, the ration of Un/Vn is...

hello, I'm working on a little puzzle and part of it requires summing the infinite series 1/(k^1.5) which clearly converges, but I've never been very good at actually finding what a series converges to. Could you give me a good swift kick in the head. Just a hint will do. Thanks,

I have a problem with an inequality. In the numberator of one term I have X sub k, and in the denominator I have the sum X sub ks from 1 to n. So let's say I use n=2 and have two terms in the denominator Xsub1 and Xsub2. What am I using for the X sub k in the numerator. It definitely is not X sub n.

I have just made the following variable switch: \sum_{i=0}^n\sum_{j=0}^m\binom{n}{i}\binom{m}{ j}x^{i+j}=\sum_{k=0}^{n+m}\sum_{i=0}^k\binom{n}{i}\binom{m}{k-i}x^{k} I know it's right, but is there a method I can use to prove without a shadow of a doubt that it is?

Hi guys, I know this may sound so "newbieish", but I really need some clarification. While resaerching over the net I came across a proof on a derivation of the Matrix p-norms. While reading, I stumbled upon this part of the proof: \| Ax \|_1 \leq \sum^n_{i=1} \left| \sum^n_{k=1}...

\sum_{x =1}^{\infty} x (\frac{1}{2})^x is there a formula for this, like for infinite geometric summation?

\sum_{p\leq N}\frac{1}{p}=\log\log N + A + O(\frac{1}{\log N}) Does it mean that we can simply replace the O part with a function that is a constant times 1/(log N)? What would be the difference between A + O(\frac{1}{\log N}) and O(1)?

HELP: A summation question Hi Given the sum \sum _{p=0} ^{\infty} (-1)^p \frac{4p+1}{4^p} I have tried something please tell if I'm on the right track Looking at the alternating series test (a) 1/(4^{p+1}) < (1/(4^p)) (b) \mathop {\lim }\limits_{p \to \infty } b_p =...

This stuff is making me bang my head against the wall. I understand the concept and notation of summation with no problems. It seems though for about every one problem I get right there is five I get wrong. The only thing I can think I'm doing wrong is bad algebra habits or I'm using the...

Hi, can someone please tell me whether or not I can switch the 'order' of the indices over which a double sum is taken? To clarify, my question is whether or not the following is true. \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\left( {a_i b_j } \right)} } \mathop = \limits^...

I can't seem to get these power series to match up so that I can solve the equation...heres my work:

Hello. I've searched around a bit for a math forum where I could get help with this and this seems like the one I found where I could get some help with this. I was posed the following problem. Now I must admit it is over my head (as is most of the math on this forum) I was hoping that...

I am to show that... \sum_{n=-N}^{+N} cos(\alpha -nx)=cos\alpha \frac{sin(N+0.5)x}{sin(x/2)} \sum_{n=-N}^{+N} cos(\alpha)cos(nx)+\sum_{n=-N}^+Nsin(\alpha)\frac{sin(N+0.5)x}{sin(x/2)} \sum_{n=-N}^{+N}sin(\alpha)\frac{sin(N+0.5)x}{sin(x/2)} =0 cos(\alpha) 2 \sum_{n=0}^{+N} cos(nx) I...

i have to prove that: n sum [C(n,k)]^2=C(2n,n) k=0 i have in my text a hint that i need to use: (1+x)^n(1+x)^n=(1+x)^2n but i got that: n 2n sum [C(n,k)]^2= sum [C(2n,k)] k=0 k=0 how do i get out of this mess?

As I don't know how to use this latex coding here it goes... if I represent by E the sum of terms where k=1 and n is the unknown I need to use the formulae for Ek and Ek^2 to obtain a formula for Ek(k+1), by simplifying the algebra as much as possible. Can someone help with this please?

Given U^k_i, the components of U is a delta function i.e for i=k U^i_k =1, to prove it is invariant under Lorentz transformation~~ I don't know how to express it in Einstein summation notation, I am very confused with the upper-lower index, is it right to write the transformation in this...

Prove that summation of n(n+1)/2 is true for all integers. Why is my proof not valid? Could someone explain to me how this is not a valid proof of the summation of "i" from i=1 to n: n(n+1)/2 Show for base cases: n=1: 1(1+1)/2=1 n=2: 2(2+1)/2=3 n=3: 3(3+1)/2=6 ... inductive...

Hello all, I am trying to prove that the following is true: lim_{M \rightarrow \infty} \sum_{P = (\frac{1}{N}-\delta)M}^{(\frac{1}{N}+\delta)M} \frac{(N-1)^{{M-P}}M!}{P!(M-P)!N^{M}} \rightarrow 1 where P , M , and N are integers, and \delta is an arbitrarily small positive...

2 math questions (summation, mathematical induction) I have 2 questions regarding summation and mathematical induction 2. Prove by mathematical induction \sum^n_{r=1} \frac {1}{r(r+2)} = \frac {3}{4} - \frac {(2n+3)}{2(n+1)(n+2)} i am now trying to prove that 3/4 -...

If \sum^{n}_{r=1} u_r =3n^2 +4n , what is \sum^{n-1}_{r=1}u_r ? I know that \sum^{n-1}_{r=1}u_r is equals to \sum^{n}_{r=1} u_r =3n^2 +4n - u_n but the answer given is 3n^2-2n-1. How do i express it in that way? thanks alot.

Help With Probability Question i have been working on this for a week can anyone help? NOTE: Look at reply from moodoo for proper matematical symbols! I need the probability of being dealt a bridge hand with at least 5 hearts. I have to possible answers but I have never done this...

The set S(N) of all natural numbers is generally believed to have infinite cardinality (ie S(N) has an infinite number of members) and yet every member of the set is believed to be finite. Infinite natural numbers are by convention "not allowed". This leads to a contradiction, as follows ...

I need to write the following series in summation notation 1) 1+3+5+7+9+11 SUMMAND (2k-1)? is this right? 2) 4+6+8+10+12+12+16+18 (2k+2)? is this right? Have I got it?

Hey. This has been bugging me for a long time: why does summation from n=1 to infinity of (-1)^n or i^n or 1/n or -1/n not converge, because summation from n=1 to infinity of 1/n^2 conveges. Don't the terms in (-1)^n or i^n or 1/n or -1/n(-1)^n tend to 0?

Can the summation... Can the summation of a number equal that numbers square root??

From http://mathworld.wolfram.com i see that the integral notation was "the symbol was invented by Leibniz and chosen to be a stylized script "S" to stand for 'summation'. " So from that i figure integrals are just summations. So what's the difference from Einstein Summation, where "repeated...

I have a rather simple question, but my rusty brain needs a good, swift kick-start. I start with: \sum_{i=1}^k i and substitute in i=k-j to get: \sum_{k-j=1}^k (k-j) How do I get from this to the following? \sum_{k-j=1}^k (k-j) \rightarrow \sum_{j=0}^{k-1} (k-j) Thanks...