Sum Definition and 1000 Threads

Homework Statement Prove that \sum[(x_{i} - \overline{x})(y_{i} - \overline{y})] = \sum[(x_{i} - \overline{x})y_{i}] Homework Equations None. The Attempt at a Solution I tried using the fact that the sum of the mean values is just the mean value, because the sum of a constant...

I have an experiment in which I want to extract the distribution function of a process. I expect it to be Gaussian. Each data point measured is an entire distribution, f(x), but I am forced to average over many points such that the result of the experiment is the sum of many measurements of...

Homework Statement Let ##X_k## be iid uniform discrete on ##\{0,...,9\}##. Find the distribution of ##\sum\limits_{k=1}^{\infty} \frac{X_k}{10^k}##Homework Equations The Attempt at a Solution I've tried a lot of things, I've tried decomposing ##X_k## into 10 bernoulli trials, I've tried using...

How does the spin of a pair of particles work if both particles are known to be chiral? generically if I sum the spins of two different (EDIT: spin 1/2, indeed ;-) particles I expect to get a triplet with S=1 \uparrow\uparrow, \uparrow\downarrow+\downarrow\uparrow, \downarrow\downarrow and...

the problem: In how many ways can we write the number 4 as the sum of 5 non-negative integers?I realize this is a generalized combinations problem. I can plug it in using a formula, but I want to understand the logic behind why the generalizaed combination formula works. More specifically, my...

Can someone guide me with the steps to differentiate a geometric sum, x? ^{n}_{i=0}\sumx^{i}=\frac{1-x^{n+i}}{1-x} If I'm not wrong, the summation means: = x^0 + x^1 + x^2 + x^3 + ... + n^i Problem is: I have basic knowledge on differentiating a normal numbers but how do I apply...

Homework Statement V(t) = [2t - 4] u(t) u(4 - t) I need to express this a sum of step and ramp functions. Homework Equations The Attempt at a Solution I have absolutely no idea how to proceed

I ran into this problem, and would like to see if there is something more elegant. Suppose we have a sequence $a_1, a_2, \dotsc, a_n, \dotsc$ where $a_k$ is the (running) sum of rolling a standard 6-side die $k$ times. E.g. What's the chance of saying the number $2$ appears in this sequence...

Equate the limit $$\lim_{n \to \infty} \frac1{n} \sum_{i = 1}^n \sum_{j = 1}^n \frac1{i + j}$$ Note : This was a challenge from a user in mathstackexchange. From a glance, there should be many ways to do it, so partly I posed this problem to see how the resident analysts in MHB handle it...

Find a positive number such that the sum of the number and its reciprocal is as small as possible. Full marks for proving your answer is correct. Process: let $a>0$ $$f(a)=a+\frac{1}{a}$$ $$f'(a)=1-\frac{1}{a^2}=\frac{(a+1)(a-1)}{a^2}$$ The critical numbers are $0$, $\pm 1$, but only $1$ is...

Find the sum of all real numbers $a$ such that $5a^4-10a^3+10a^2-5a-11=0$.

Hiya. I got to an interesting bit in a calculus book, but as usual I'm stumped by a (probably simple) algebraic step. The author goes from: (ds)^2=(dx)^2+(dy)^2 to: ds=\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx I understand moving the square root across, but I don't understand how the...

Hi. Assume there's a probability ##q## for a guy to take a step to the right, and ##p=1-q## to take one to the left. Then the probability to take ##n## steps to the right out of ##N## trials is ##P(n) = {{N}\choose{n} }q^n p^{N-n}##. Now, what is ##<n>##? My textbook in statistical physics...

Find an integer $k$ for which $\dfrac{1}{k}+\dfrac{1}{k+1}+\dfrac{1}{k+2}+\cdots+\dfrac{1}{k^2}>2000$.

Homework Statement I'm doing question 23 in Chapter 4 of Spivak's Calculus on Manifolds. The question asks, For R > O, and n an integer, define the singular l-cube, c_{R,n} :[0,1] \rightarrow \mathbb {R}^2 - 0 by c_{R,n} (t) = (Rcos2\pi nt, Rsin2\pi nt). Show that there is a singular...

Suppose $X$ is a number of the form $\displaystyle X=\sum_{k=1}^{60} \epsilon_k \cdot k^{k^k}$, where each $\epsilon_k$ is either 1 or -1. Prove that $X$ is not the fifth power of any integer.

help! find the magnitude of the resultant force and the angle it makes with the positive x-axis. i don't have any examples in my book like this one

I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding Theorem 2.31 (regarding the direct sum of n vector spaces) on pages 61-62. Theorem 2.31 and its accompanying text...

I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding the Universal Mapping Property of direct sums of vector spaces as dealt with by Knapp of pages 60-61. I am not...

$\sum_{n=0}^{\infty}\frac{2^n}{3^nn!}$ is this correct? $\sum_{n=0}^{\infty}(\frac{2}{3})^n \frac{1}{n!}$ $\sum_{n=0}^{\infty}\frac{(x)^n}{n!}=e^x$ $x=2/3$ $e^x=e^{2/3}

Homework Statement How to get from Sum of 2(cos((3pi)/(2^(k+1)))sin(pi/(2^(k+1)))) from k = 1 to infinity to Sum of sin((4pi)/(2^(k+1))) - sin((2pi)/(2^(k+1))) from k = 1 to infinity The two expressions are equivalent. I need help getting from the first expression to the second.

For positive integers $p,\,q,\,r,\,s$ such that $ps=q^2+qr+r^2$, prove that $p^2+q^2+r^2+s^2$ is a composite number.

Prove the Sum Rule for Limits $$\lim_{x\to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) = L + M$$ Proof Assume the following: $$\lim_{x \to a} f(x) = L, \space\lim_{x \to a} g(x) = M$$ Then, by definition ##\forall \epsilon_1 > 0, \exists \delta_1 > 0## such that...

how many term of the series $\sum_{n=2}^{\infty}\frac{1}{[n(ln (n))^2]}$ would you need to add to find its sum to within 0.01? approximate the sum of the series correct to four decimal places. $\sum_{n=1}^{\infty}\frac{(-1)^n}{3^nn!}$

Homework Statement Sum starting from n=1 to infinity for the expression, (3/4^(n-2)) What the solutions manual has done is multiply the numerator and the denominator by 4. 12/(4^(n-1)) I don't know what they have done from here on: 12 / (1 - 1/4) = 16 Can someone...

Hey guys, I just wanted to run a quick series question by you guys just to confirm my answer. I'm doubting whether or not I should keep going or if S6 is enough. I got S5 = -0.28347 and S6 = -0.28347, so that is where I concluded than Sn ~ -0.2835. I would appreciate it if someone could...

I have this sum $$\left(N+1\right)^{2}\underset{j=1}{\overset{N}{\sum}}\frac{\left(-1\right)^{j}}{2j+1}\dbinom{N}{j}\dbinom{N+j}{j-1}\underset{i=1}{\overset{N}{\sum}}\frac{\left(-1\right)^{i}}{\left(2i+1\right)\left(i+j\right)}\dbinom{N}{i}\dbinom{N+i}{i-1}$$ and numerical test indicates that is...

The definition (taken from Robert Gilmore's: Lie groups, Lie algebras, and some of their applications): We have two vector spaces V_1 and V_2 with bases \{e_i\} and \{f_i\}. A basis for the direct product space V_1\otimes V_2 can be taken as \{e_i\otimes f_j\}. So an element w of this space...

Find the sum of all real solutions for $x$ to the equation $\large (x^2+4x+6)^{{(x^2+4x+6)}^{(x^2+4x+6)}}=2014$. P.S. I know this doesn't count as a challenge(no matter how you slice it) because it's quite obvious and rather a very straightforward sort of problem but I'd like to share it...

Let $PQRST$ be a pentagon inscribed in a circle such that $PQ=RS=3$, $QR=ST=10$, and $PT=14$. The sum of the lengths of all diagonals of $PQRST$ equals to $\dfrac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.

Homework Statement prove by induction \sum_{j=1}^{n+1} j \cdot 2^j = n \cdot 2^{n+2}+2; n \ge 02. The attempt at a solution P(0) \sum_{j=1}^{0+1} j \cdot 2^j = 0 \cdot 2^{0+2}+2 2+2 here is where I need some help is P(k) \sum_{j=1}^{k+1} j \cdot 2^j = (k+1) \cdot 2^{k+3}+2 ?? then...

Express $\displaystyle \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \dfrac{1}{m^2n+mn^2+2mn}$ as a rational number.

1^3+2^3+...+n^3 = \left[ \frac{n(n+1)}{2}\right]^2; n\ge 1 P(1) = 1^3 = \frac{8}{8} = 1 P(k) = 1^3+...+k^3 = \left[ \frac{k(k+1)}{2}\right]^2 (induction hypothesis) P(k+1) = 1^3+...+k^3+(k+1)^3 = \left[\frac{(k+1)(k+2)}{2}\right]^2 I start getting stuck here I foiled it out then let m =...

I have a series in Prabhakar three parametric mittag-leffler is there any article on that

an object is moving in the direction i + j is being acted upon by the force vector 2i + j, express this force as the sum of a force in the direction of motion and a force perpendicular to the direction of motion. the parallel would be \hat{i}+\hat{j} and the orthogonal would be \hat{i} -...

How does one prove the following: \int^{c}_{a} f\left(x\right)dx = \int^{b}_{a} f\left(x\right)dx +\int^{c}_{b} f\left(x\right)dx where f\left(x\right) is continuous in the interval x\in \left[a, b\right], and differentiable on x\in \left(a, b\right). My approach was the following...

Find the number between 1000 and 2000 that cannot be expressed as sum of (that is >1) consecutive numbers.( To give example of sum of consecutive numbers 101 = 50 + 51 162 = 53 + 54 + 55 ) and show that it cannot be done

Problem: Let $[x]$ be the nearest integer to $x$. (For $x=n+\frac{1}{2}, n\in \mathbb{N}$, let $[x]=n$). Find the value of $$\sum_{m=1}^{\infty} \frac{1}{[\sqrt{m}]^3}$$ Attempt: I tried writing down a few terms and saw that $1$ repeats $2$ times, $2$ repeats $4$ times but I didn't check it...

Homework Statement Find the expectation value of the Energy the Old Fashioned way from example 2.2. Homework Equations ##\left< E \right> =\frac { 480\hbar ^{ 2 } }{ \pi ^{ 4 }ma^{ 2 } } \sum _{ odds }^{ \infty }{ \frac { 1 }{ { n }^{ 4 } } } ## The Attempt at a Solution Never...

If $p,\,q,\,r,\,s$ are positive integers with sum 63, what is the maximum value of $pq+qr+rs$?

Homework Statement Recognize the series $$3-3^3/3!+3^5/5!-3^7/7!$$ is a taylor series evaluated at a particular value of x. Find the sumHomework Equations Sum of Infinite series = ##a/1-x## The Attempt at a Solution So, I can't figure out what i would us as the ratio (the thing you multiply...

The gist of the approach I took is that∑1/p = log(e^∑1/p) = log(∏e^1/p) and logx→ ∞ as x→∞. Proof outline: let ∑1/p = s(x). (...SO I can write this easily on tablet) and note that e^s(x) diverges since e^1/p > 1 for any p and the infinite product where every term exceeds 1 is divergent. Then...

How can we prove $$\displaystyle \tan^{-1}\left(\frac{4}{7}\right)+\tan^{-1}\left(\frac{4}{19}\right)+\tan^{-1}\left(\frac{4}{39}\right)+\tan^{-1}\left(\frac{4}{67}\right)+...\infty = \frac{\pi}{4}+\cot^{-1}(3)$$ My Trial: First we will calculate $\bf{n^{th}}$ terms of Given Series...

if $2^w+2^x+2^y+2^z=20.625$ here $w>x>y>z$ and $w,x,y,z \in Z$ find $w+x+y+z$

if $(4^m+4^n)$ mod 100=0 (here $m,n\in N \,\, and \,\,m>n$) please find:$min(m+n)$

Hi, I was taught that a standing wave is formed when a progressive wave meets a boundary and is reflected. I was also taught that waves that meet a fixed end, reflect on the opposite side of the axis to the side that they met it at. (I hope that makes sense) If this is true, when the wave is...

Hi everybody, I am looking for some help with a problem that has been nagging me for some time now. I'm going to give you the gist of it, but I can provide more details if needed. So, after some calculations I am left with a function of the following form $$ F_L(y) = f(y) -S_L(y)...

I am writing this in C#. Here is the code. using System; namespace ConsoleApplication3 { class Program { static void Main(string[] args) { int sum = 0; int uservalue; Int32.TryParse(Console.ReadLine(),out uservalue)...

While doing an another problem, I came across the following sum and I have no idea about how one should go about evaluating it. $$\sum_{k=0}^{\infty} (-1)^k\left(\frac{1}{(3k+2)^2}-\frac{1}{(3k+1)^2}\right)$$ Wolfram Alpha gives $-\frac{2\pi^2}{27}$ as the result but I have absolutely no idea...

Show that \sum_{j=1}^{j=n}\binom{n}{j} \frac{(-1)^{j+1}}{j} = 1 +\frac{1}{2} +\frac{1}{3} + \cdots +\frac{1}{n}