Summation Definition and 627 Threads

(1/2) + (2/4) + ... + (n/(2^n)) = sum i=1 to i=infinity of (i/(2^i))?i know how to express the sum of just 1/(2^i), but not the above thanks for the help!

When needed to do that, I found it much easier to pretend it's an integral summation and then draw the area diagram then work it out from the picture the new terminals for the integral. Then convert that back into the discrete sum. Is that how you would do it? However for three or more...

Given nonzero whole numbers n, prove 13+23+33+...+n3=(1+2+...n)2 I figured this out numerically, but lack the skills to solve it analytically (no doubt by induction) and could not find it in my table of summations. I'm too old for this to be homework.

Can someone help me break this down? \Sigma^{k}_{i=1}\frac{i \left(^{n}_{i}\right)\left(^{m}_{k-i}\right)}{\left(^{m+n}_{k}\right)}

I was looking at the web page containing a derivation for the Poisson distribution: http://en.wikipedia.org/wiki/Poisson_distribution which derives it as the limiting case of the binomial distribution. There is a simplification step which I am missing, which is the step(s) between...

Im not sure if it is related to calculus but, Calculate the sum \sum^{\infty}_{n=0}\frac{(n-1)(n+1)}{n!} exactly. I tried to to partial fraction decomposition but couldn't find anything.

I have the summation x(1/2)^x for (x=1,2,3,4,...) So I set it up as s=1(1/2)+2(1/4)+3(1/8)+4(1/16)... This is however where I'm lost, I'm not exactly sure how to sum an infinite sequence, it hasn't really been introduced in any of my math courses, it just popped up in a statistics problem...

Getting E[N] from the multinomial dist, where \frac{n!}{n_{1}!n_{2}!... n_{r}!}p^{n_{1}}_{1}}p^{n_{2}}_{2} ... p^{n_{r}}_{r} is the pmf. Does this look right? \Sigma^{n}_{i=1}E\left[e^\left\{{\Sigma^{r}_{k=1}t_{k}N_{k}}\right\}}\right]...

Has anyone heard about a way to find the sum of a serie of this form: s=\sum_i{\exp(a+b\sqrt(i))}

[SOLVED] Summation Notation - Variable in the exponent Homework Statement This is an example formula. How do I solve a summation if something is in the form of Sum(c^i), where c is some constant?

[SOLVED] Summation - Riemann Intergral - URGENT Homework Statement Im working on the upper and lower riemann sums of f(x) = exp(-x) where Pn donates the partition of [0,1] into n subintervals of equal length (so that Pn = {0,1/n,2/n,...,1}) Homework Equations The Attempt at...

I'm having trouble picking apart this summation: \sum^{inf}_{n=1} P(E)*P(1-p)^{n-1}; where p = P(E) + P(F) I know I need to use the identity of a geometrical series when |r| < 1 : 1/(1-r) I'm getting P(E)/(1-(P(E)+P(F)) But I need to be getting P(E)/((P(E)+P(F)); The entire...

Homework Statement I am looking for a closed form of the summation: sin(x) + sin(3x) + sin(5x) + ... + sin((2n-1*)x) Homework Equations None. The Attempt at a Solution Through a complete stroke of luck, I believe I have arrived at the correct solution: sin^2(nx)/sin(x) I have...

[SOLVED] radius of convergence of an infinite summation Homework Statement find the radius of convergence of the series: \sum\frac{(-1)^k}{k^2 3^k}(z-\frac{1}{2})^{2k} Homework Equations the radius of convergence of a power series is given by \rho=\frac{1}{limsup |c_k|^{1/k}}...

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} ? (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...

y(t) = \sum C_{n} e^{-\gamma t} sin(n \omega t) y(t) is a summation of a large number of arbitrarily decaying sinusoids with arbitrary coefficients. Find the value of t at which y(t) is a maximum. Personally, I have doubts that this can be solved without knowing what the constants and...

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...