Sum Definition and 1000 Threads

$\sum\limits_{n = 1}^{\infty}\left(\sqrt{1 + n^2} - n\right)$ $$ \sqrt{1 + n^2} - n = \frac{1}{\sqrt{1 + n^2} + n} $$ Now what?

$\sum\limits_{n = 2}^{\infty}n^p\left(\frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}}\right)$ where p is any fixed real number. If this was just the telescoping series or the p-series, this wouldn't be a problem.

Calculate 5 \Sigma r=0 r(r+1) (Sorry I don't know how to do the proper notation online) How do you calculate the sum for this since the common difference is changing? I tried to write them out separately so the sum of r x sum of r+1 but I don't know how you put that in the sum formula. Sn =...

Homework Statement I have this series 1^{3}-2^{3}+3^{3}-4^{3}+5^{3}-6^{3} + \ldots Homework Equations and sequence of partial sums for this series that is: S_n = \sum_{k=0}^{n}(-1)^{k+1} k^3 = \dfrac{1 + (-1)^n(4n^3 + 6n^2-1)}8 =\begin{cases} \dfrac{2n^3+3n^2}4; & n \text{ is...

Homework Statement Let V be a real inner product space, and let v1, v2, ... , vk be a set of orthonormal vectors. Prove Ʃ (from j=1 to k)|<x,vj><y,vj>| ≤ ||x|| ||y|| When is there equality? Homework Equations The Attempt at a Solution I've tried using the two inequalities given to us in...

Homework Statement Find the sum of the sequence: 2, -2/3, 2/9, -2/27, 2/81, . . . Homework Equations The Attempt at a Solution I can see that the number is multiplied by -1/3, but I'm unsure of how to find the sum. Any pointers?

Why oh WHY do the arrows in this TORQUE problem keep alternating between SINE and COS This is what they did: I understand the whole Torque = F * r, but WHY (at the top of the diagram) is the arrow pointing up = (25 N)cos 30 and NOT sin and why is the arrow pointing to the right...

Lagrange's four-square theorem states that any natural number can be expressed as the sum of four integer squares. I've noticed that the first few values of 8n-1 can all only be expressed as a minimum of the sum of four squares. Is this true for all values of n? What's the proof behind it? Thanks.

Hi, I have determined, correctly I believe, that the following series converges: 1/[(3n-2)(3n+1)] Now I am asked to determine its sum. I have tried separating it into two subseries, but each time got a p-series with p=1, hence to no avail. The answer should be 1/3, but how may it be...

I teach cost-benefit analysis, which requires me to teach monte carlo simulation for sensitivity analysis. I use excel. I understand how to generate a number with uniform, triangular, normal or other distributions, but I don't know how to randomly generate a set of numbers between zero and one...

Homework Statement Homework Equations I really wish they existed in my notes! *cry*. All I can think of is that integrating or in other words summing the dirac delta functions for all t, would be infinite? None the less the laplace transform exist since its asked for in the question and i...

Find the sum of \Sigma 200 r=5 5r-2 Sn = n/2 [2a + (n-1)d ]I used S 200 and I got about 101400 but then when I verified on my calculator it was 100058, my calculator has the sigma notation for working out the sum of , how do you get 100058?

i'm trying to prove the sum of nth roots of unity = 0, but I don't really know how to proceed: suppose z^n = 1 where z ε ℂ, suppose the roots of unity for z are 1, ω, ω^2, ω^3 ... ω^n the sum of these would be S = 1 + ω, ω^w, ω^3 +...+ ω^(n-1) + ω^n from here I had an idea to do some...

Homework Statement Estimate the area of the region under the curve y = ln(x) for 1 ≤ x ≤ 5. Use the left-hand rule with n = 50. Homework Equations The Attempt at a Solution Do I really have to do 50 calculations? There has to be a faster way :/ (aside from using the definite...

Homework Statement Show that \sum_{k=0}^{\infty} \sqrt[k]k-1 converges. Homework Equations Ratio, radix theorems, comparison with other sums... The Attempt at a Solution No idea whatsoever. Where does one begin in this case ? With other cases I'm quite confident.

Does there exist a sequence of real nonzero numbers whose sum converges to 0? I would think there isn't, but I'm interested in people's opinions and arguments. For any nonzero m, a series of nonzero numbers whose sum converges to m can easily be constructed using the formula: \sum...

My answer is almost correct, except for the negative sign. Can anyone help? Many thanks. Homework Statement Q. Without using a calculator, show that \sin10^{\circ}+\sin80^{\circ}=\sqrt{2}\cos35^o Homework Equations The Attempt at a Solution...

Homework Statement I am trying to prove how \(g''(r)=\sum\limits_{k=2}^\infty ak(k-1)r^{k-2}=0+0+2a+6ar+\cdots=\dfrac{2a}{(1-r)^3}=2a(1-r)^{-3}\). I don't know what I am doing wrong and am at my wits end. The Attempt at a Solution (The index of the summation is always k=2 to infinity)...

Homework Statement Let X,W,Y be iid with a common geometric density f_x(x)= p(1-p)^x for x nonnegative integer and p is in the interval (0,1) What is the characteristic function of A= X-2W+3Y ? Determine the family of the conditional distribution of X given X+W? Homework Equations...

Homework Statement The notation for a Riemann sum - Ʃ f(x*i)Δx - is very similar to the notation for the integral (the Ʃ becomes ∫, the f(x*i) becomes f(x) and the Δx becomes dx). \int f(x)dx = \lim_{n \to \infty}\sum_{k=0}^{n} f(x_i) Δx Is there a way to explicitly define the values on the...

Homework Statement let y_1 and y_2 be iid bin(5,1/4) random variables let v=y_1+2*y_2 and u = 3*y_1 -2y_2 find f_uv (u,v) and the cov(u,v) Homework Equations f_y (y) = (5 choose y) (1/4)^y (3/4)^5-x for x=0,1,2,3,4,5 covariance=E(uv)-E(u)E(v) The Attempt at a Solution...

$$ 2\frac{1}{3} + \frac{1}{3^2} + 2\frac{1}{3^3} + \frac{1}{3^4} + 2\frac{1}{3^5} + \cdots = \frac{1}{3}\sum_{n = 0}^{\infty}\left(\frac{1}{3}\right)^n $$ I am stuck on what to add into account for the 2 at every other term.

Homework Statement Please write a specific function to define this series. Also provide a sum that the series converges to.Homework Equations Sn - {1, 1+1/e2, 1+1/e2+1/e4, 1+1/e2+1/e4+1/e6, ...} The Attempt at a Solution I know that the common ratio is 1/e2 and that you can raise that to...

f(x,y) = (1/x) for 0≤y≤x≤1 A new rv Z=X+Y where X,Y not independent find the pdf of z My approach F(z) = P(Z≤z) = ∫∫fXY(x,y) dx dy x= -∞ to ∞ y= 0 to z-y f(z) = d/dz(F(z)) = ∫fXY(z-y,y) dy y= -∞ to ∞ (using Leibnitz) where i am stuck is this doesn't converge

Homework Statement Does this converge or diverge? Ʃ1/(1+2+3+4+5...+n), as n---> infinity?The Attempt at a Solution I rewrote this into Ʃ(Ʃ1/n) (is it correct?). I figured that since Ʃ(1/n) diverges, then the sum of each partial sum most (obviously) also diverge. However, it appears I'm...

Let $S=\{ 1, 2, \ldots , 2p\}$, where $p$ is an odd prime. Find the number of $p$-element subsets of $S$ the sum of whose elements is divisible by $p$.Attempt. Let $\mathcal{K}$ be the set of all the $p$ element subsets of $S$. Let $\sigma(K)$ denote the sum of the elements of a member $K$ of...

Homework Statement The division of a polynomial f(x) by (x – 1)(x – 2) has remainder x + 1. If the remainder of the division of f(x) by (x – 1) and (x – 2) are, respectively, a and b. Then what is a^2 + b^2? Homework Equations I guess the remainder theorem could be useful here. The...

1/5 + 1/7 + ..... + 1/401

Z is the set of integers. Prove that (3/7)Z + (11/2)Z = (1/14)Z Attempt: By definition, (3/7)Z+(11/2)Z={3k/7 + 11m/2 : k,m € Z} = {(6k + 77m)/14 : k,m € Z}. Showing that 3/7Z+11/2Z is a subset of 1/14 Z is easy but I can't prove the converse. Can't show that whatever n€1/14Z I take...

Homework Statement Find the derivative of y = x^2 + x^(2x)The Attempt at a Solution By looking at the equation I think I need to use implicit differentiation + natural logs. But I can't do anything with: lny = ln(x^2 + x^(2x)) So I assume I'm wrong.. Any help??

Homework Statement Here is the problem: Here is the solution: I am confused about part c. The Attempt at a Solution Here is my attempt at part c: I don't get why the answer key solution sums the moments about point G. I decided to sum up the moments about C and D and...

∞ Ʃ 1/(n(n+8)) n=1 So i used partial fractions and got (1/8)/(n) - (1/8)/(n+8) From there i pulled out the 1/8 so now my equation is ∞ (1/8) Ʃ (1/n)-(1/(n+8)) n=1 So from here do i just start doing like s1= (1/8)(1-1/9), s2=(1/8)(1/2-1/10) to find...

Hi, Is there a general algebraic expression for the sum of a function inside a radical? I mean for something like this? \sum^{n}_{i=1}\sqrt{f(i)} The specific case is given with constant c: \sum^{n}_{i=1}\sqrt{c^4i^4+c^2i^2+1} And I supposed a related question is that, is there some way of...

Hi, I'm writing a program in fortran that basically creates a loop from 1 until a certain number x (input by the user), and goes through each value between 1 and the certain number x in order to determine if each value meets certain criteria. The criteria is that the sum of the square of the...

I was working on some algebraic topology matters, thinkgs like the connected sum of some surfaces is some other surface. And for this study, I was using the Munkres's famous textbook 'Topology' the algebraic topology part. My qeustions are as follows: Q1) Munkres introduces 'labelling scheme'...

Hi everyone. I have learned that: 1+2+3+...=\frac{n(n+1)}{2} 12+22+32=\frac{n(n+1)(2n+1)}{6} I want to know what the general formula of Ʃna, in which n and a are natural numbers, respect to n and a.

From a fraction with infinite sum in denominator to partial fractions?? I am currently studying a course on Perturbation Methods and in particular an example considering the following integral \int_{0}^{\frac{\pi}{4}} \frac{d\theta}{\epsilon^2 + \sin^2 \theta}. There's a section of the...

S = \frac{1}{2} + \frac{1}{4} + ... + (\frac{1}{2^n}) I noticed that this is a sum of a infinite series with the common ratio being 1/2, so using \frac{1}{1-1/2} I get S = 2, however with this question there is a hint saying multiply S by 2, which I did not use so I'm worrying if I done...

The drawing show a barometer with water inside. If I move up object with air inside I recover PV energy (P=external pressure, V=volume of object). If I want to move out object when it is at top, I need PV, is that ? So if it's that. If I replace: 1/ gravity by balls attracted with springs (for...

Hi, I need your help, Say we have two random variables with some joint pdf f(x,y). How would I go about finding the pdf of their sum?

I need to calculate \sum_{n=0}^{∞}x^{(2^n)} for 0≤x<1. It doesn't resemble any basic taylor series, so I have no idea how to sum it up. Any hint, or the resulting formula? This series comes from a physical problem, so I suppose (if I didn't make a mistake) that the series is sumable, and...

Homework Statement If the weight of the boom is negligible compared with the applied 45-kN load, determine the cable tensions T1 and T2 and the force acting at the ball joint at A. Homework Equations The Attempt at a Solution I have successfully calculated the tensions in...

Provided is a function f(x)=\sum_{j=1}^n ||x-x_j||, for x being a two dimensional vector, where ||.|| denotes the Euclidean distance in 2D space. How could one obtain a derivative of such a function?

Having a lot of trouble with this question. So first I tried making an equation, and I wrote that the probability = P(rolling a 6)+P(rolling a 6 and not a 7 on the first roll and not a 9 on the first roll) + P(rolling a 6 and not a 7 on the first roll and not a 9 on the first roll and not a 7...

I know how to differentiate the dot and cross products of two vectors, is the differentiation of a vector sum done like this: d/dt (u+t) = u' + t + u + t' Or simply add them and then differentiate? Thanks

Homework Statement \sum_{n=2}^{\infty} \frac{1}{n^2-1} Homework Equations The Attempt at a Solution After doing partial fraction decomposition, I discovered that it was a telescoping series of some sort; the partial sum being 1/2[ (1 -1/3) + (1/2 - 1/4) + (1/3 - 1/4) +...] The...

Homework Statement I need to prove both of these (in exercise 11) http://postimage.org/image/x7shxv11f/ Homework Equations The dot product The Attempt at a Solution

Homework Statement If the magnitude of the sum of two vectors is less than the magnitude of either vector, then: -the vectors must be parallel and in the same direction -the scalar product of the vectors must be negative -none of these -the scalar product of the vectors must be...

Homework Statement h(x)=\left\{\begin{matrix} 9+2x , 0<x<\pi\\ -9+2x , -pi<x<0 \end{matrix}\right. \\ Find \ the \ sum \ of \ the \ Fourier \ series \ for \ x=\frac{3\pi}{2} and\ x=\pi \\ The \ Fourier \ series \ is: \\ h(x)=9+\pi + \sum_{n=1}^{inf}...

http://books.google.rs/books?id=vrcHC9XoHbsC&pg=PA234&lpg=PA234&dq=Quantum+theory+of+magnetism+Ising&source=bl&ots=5uRLh1gzEf&sig=ZNHUXGgzbDIW4nHy3Txdmi4mGb8&hl=sr&sa=X&ei=YORpUN6ZO7HN4QTwnoDgAw&ved=0CEsQ6AEwBA#v=onepage&q=Quantum%20theory%20of%20magnetism%20Ising&f=false In page 266 why we...