Sum Definition and 1000 Threads

Hello, I'm trying to figure out the free body diagrams for the inverted pendulum problem and I'm having trouble figuring out the one equation: ##Psin\theta + Ncos\theta - mgsin\theta = ml\ddot{\theta} + m\ddot{x}cos\theta## I've never really seen a mixed sum of forces equation before where some...

Homework Statement How to use binomial theorem for finding sums with binomial coefficients? Example: S={n\choose 1}-3{n\choose 3}+9{n\choose 5}-... How to represent this sum using \sum\limits notation (with binomial theorem)? Homework Equations (a+b)^n=\sum\limits_{k=0}^{n}{n\choose...

Hi! If I have a sequence that its first 4 terms are: 30, -31, +32, -32 The pattern is geometric sequence but has alternating signs.. How can I find its sum .. I know it is composed of 2 sequences .. However, when I try to separate the 2 sequences .. I get them of different "lengths" In...

Homework Statement [/B] Calculate \widehat{Y^{2}} (i.e., the mean of the square of Y. Homework Equations Y=\sum_{k=0}^{N-1}y_{k} where y_{k}=e^{-\gamma t}e^{\gamma \tau k}G_{k} and t=N\tau The quantities y_{k} (or G_{k}) are statistically independent. The Attempt at a Solution...

If R = [−3, 1] × [−2, 0], Use a Riemann sum with m = 4, n = 2 to estimate the value of ∫∫R(y2 − 2x2) dA. Take the sample points to be the upper left corners of the squares. So far, I found the indefinite integral of the function to be y3/3 - 2x3/3 Not sure where to go from here

I appreciate the opportunity afforded by this forum to submit a question. I have struggled with the derivation shown in the attached picture. I am certainly unfamiliar with the concept used to include the arctan function in the encircled step. Would be highly appreciative of a prompt.wirefree

I am trying to understand a condition for a nonincreasing sequence to converge when summed over its prime indices. The claim is that, given a_n a nonincreasing sequence of positive numbers, then \sum_{p}a_p converges if and only if \sum_{n=2}^{\infty}\frac{a_n}{\log(n)} converges. I have tried...

Homework Statement Figure out the minimum sum of products for g(r s t) = r't' + rs' +rs 2. The attempt at a solution I understand you can simplify it with the Boolean theorems (e.g r't' + r = t' + r) , however how would you solve it using K-maps? I drew out a truth table, but it seems as if...

Let $x=\sqrt{7}+\sqrt{5}-\sqrt{3},\,y=\sqrt{7}-\sqrt{5}+\sqrt{3},\,z=-\sqrt{7}+\sqrt{5}+\sqrt{3}$. Evaluate $\dfrac{x^4}{(x-y)(x-z)}+\dfrac{y^4}{(y-z)(y-x)}+\dfrac{z^4}{(z-x)(z-y)}$.

Homework Statement Find an example of two unitary matrices that when summed together are not unitary. Homework EquationsThe Attempt at a Solution A = \begin{pmatrix} 0 & -i\\ i & 0\\ \end{pmatrix} B = \begin{pmatrix} 0 & 1\\ 1 & 0\\ \end{pmatrix} A+B = A = \begin{pmatrix} 0 & 1-i\\ 1+i &...

Homework Statement Show that the sum of two nxn Hermitian matrices is Hermitian.Homework Equations Hermitian conjugate means that you take the complex conjugate of the elements and transpose the matrix. I will denote it with a †. I will denote the complex conjugate with a *. The Attempt at a...

Hi! I've tried a couple of approaches with this: converting to complex exponential form and using standard trigonometric identities but have been unable to solve. I suspect DeMoivre's formula applies but I don't see how.Prove...

Consider the two divergent series: $$\sum_{n=k}^{\infty} a_n$$ $$\sum_{n=k}^{\infty} b_n$$ Is it possible for ##\sum_{n=k}^{\infty} (a_n \pm b_n)## to converge?

Homework Statement Hi all! I have been blindly drawing normal forces till today and i stumbled on this question. Homework EquationsThe Attempt at a Solution I have drawn 3 normal forces, - Hand - Leg - Center of mass Is there a normal force at the center of mass and is it the sum of the...

I am given three sine waves with individual frequency being 10 Hz, 50 Hz, and 100 Hz. What is the frequency of the following : y(t) = sin(2π10t) + sin(2π50t) + sin(2π100t) Is it simply 100, the LCM of all the sin waves? If not, How to calculate the frequency of y(t) ?

Given two numbers: A + \sqrt{A^2 - B^2} and U + \sqrt{U^2 - V^2} OBS: A, B, U and V are real numbers. I want sum it and express the result in the same form: A + \sqrt{A^2 - B^2} + U + \sqrt{U^2 - V^2} = x + \sqrt{x^2 - y^2} So, x depends of A and U. And y depends of B and V: x = x(A, U)...

$A=\begin{Bmatrix} {1,2,3,4,5,------,2015} \end{Bmatrix}$ if we pick $n$ numbers from $A$, we call it the set $B$ ,and the sum of any three numbers from $B$ are multiple of 27 ,find $max(n)$ , and the largest number we can choose from $A$

Homework Statement 1. What is the gravitational energy (relative to the unstretched surface of the trampoline) of the 20kg ball at its apex 2.0m above the trampoline 2. What is the kinetic energy of the ball just before impacting the trampoline 3. At maximum stretch at the bottom of the motion...

$a=1,2,3,4,5,------2011$, the roots of the equations $x^2-2x-a^2-a=0,$ are : $(\alpha_1,\beta_1),(\alpha_2,\beta_2),----------,(\alpha_{2011},\beta_{2011})$ respectively please find : $\sum_{n=1}^{2011}(\dfrac{1}{\alpha_n}+\dfrac {1}{\beta_n})$

If $a,\,b$ and $c$ are positive integers such that $16a b c+4a b+4a c+4b c+ a+b+c =4561$, find the sum of $a+b+c$.

Homework Statement ABC is a triangle with I as the centre of the incircle and K as the centre of the circumcircle (I ≠ K). d(X) is the sum of the distances from a point X inside the triangle to the three sides. Verify if d(P)=d(Q) (P and Q are points inside the triangle), the lines PQ and IK...

I would like to know some history on the subject like who is the first to think about sum of squares of integers and what he/she was thinking about. I think maybe it is related to Pythagorean triples. Thanks

Prove that$\lim_{{n}\to{\infty}}\sum_{j=0}^{n} {n \choose j} \frac{{(x-a)}^{n+j}}{(n+j) !} = 0 $ thanks Sarrah

I have been looking through the book Counterexamples: From Elementary Calculus to the Beginning of Calculus and became interested in the section on periodic functions. I thought of the following question: Suppose you have a periodic real valued function f(x) with a fundamental period T. Let c...

Homework Statement This problem is not from a textbook, it is something I have been thinking about after watching some lectures on Fourier series, the Fourier transform, and the Laplace transform. Suppose you have a real valued periodic function f with fundamental period R and a real valued...

Here's the problem statement from HackerRank: https://www.hackerrank.com/contests/programaniacs-june-15/challenges/sum-of-squares-1 Since the constraints are small, I tried a DP solution. Code I have written so far: #include <cmath> #include <cstdio> #include <vector> #include <iostream>...

Homework Statement I put up the image so that you can see the hints if you're curious. I am supposed to prove that if ## S=\sum_{n=0}^{\infty}a_{n}x^{n}## converges for ##|x|<1##, and if ##|a_{n+1}|<|a_{n}|## for ##n>N##, then $$|S-\sum_{n=0}^{N}a_{n}x^{n}|<|a_{N+1}x^{N+1}|\div (1-|x|)$$...

It is well known that the below series are divergent $1 - 1 + 1 - 1 + \cdots $ $1 - 2 + 3 - 4 + \cdots $ $1 + 2 + 3 + \cdots $ But after i watched a video in youtube for the channel " Numberphile " they proved that the first is equal to 1/2 , 1/4 and the last one is -1/12 ! The way to...

Homework Statement Use the method of Problem 6 to show that ∑1≤k≤n kp can always be written in the form (np+1) / (p+1) +Anp+Bnp-1+Cnp-2+... Source: Calculus by Michael Spivak. Chapter 2 problem 7. Homework Equations The method from problem 6 is described as follows: The formula for the sum of...

Why bother writing a given integer as the sum of two squares? Does this have any practical application? Is there an introduction on a first year number theory course which would motivate students to study the conversion of a given integer to sums of two squares?

Why bother writing a given integer as the sum of two squares? Does this have any practical application? Is there an introduction on a first year number theory course which would motivate students to study the conversion of a given integer to sums of two squares?

Homework Statement Alfred puts 985 USD on his bank account every time he has a birthday. Alfred just turned 48. He started to save money when he turned 35 (including 35th birthday). How much money is there on his savings-account if the interest-rate was 3.7% every year and that he had no money...

I've seen the parameterization of a^2+b^2=c^2 and also a^2+b^2=c^2+d^2, but I don't know how they arrived at those parameterizations. Would it be possible to parameterize something with two equalities like a^2+b^2=c^2+d^2=e^2+f^2? Any help is appreciated!

Homework Statement The story is that I would like to evaluate the coinciding point limit (when ## (x^0, x^1)→(y^0,y^1)##) of these two terms: \begin{eqnarray*} &&\frac{1}{2L}e^{\frac{i}{2}eE\left((x ^1)^2-(y^1)^2\right)}\left( im\left( x^0-y^0+ x^1-y^1\right) \right)...

Hi, everybody. Mathematic handbooks have given a sum formula for the modified Bessel function of the second kind as follows I have tried to evaluate this formula. When z is a real number, it gives a result identical to that computed by the 'besselk ' function in MATLAB. However, when z is a...

I am trying this problem on CodeChef: Just a simple sum My task is to evaluate: $$\sum_{i=1}^n i^i \pmod m$$ Following is the code I have written: #include <iostream> using namespace std; typedef long long ll; ll modularPower(ll base, ll exponent, ll M) { ll res = 1; while...

From Vieta's Formulas, I got: $a=2r+k$ $b=2rk+r^2+s^2$ $65=k(r^2+s^2)$ Where $k$ is the other real zero. Then I split it into several cases: $r^2 + s^2 = 1, 5, 13, 65$ then: For case 1: $r = \{2, -2, 1, -1 \}$ $\sum a = 2(\sum r) + k \implies a = 13$ Then for case 2: $r^2 + s^2 = 13$, it...

Homework Statement Write the Maclaurin series for ##\frac{1}{(1+x)^{1/2}} ## in ##\sum## form using the binomial coefficient notation. Then find a formula for the binomial coefficients in terms of n. Homework Equations 3. The Attempt at a Solution [/B]...

I am really not sure about this ... I may be generalizing it ...but anyway... suppose we have two resistors in series...connected by ofcourse a voltage source...and a certain amount of total current is flowing through the whole circuit.. now if we want to make the total current totally...

Homework Statement Consider ##\sum\limits_{n=0}^{\infty} \frac{n+1}{(2n)!}(x+1)^{2n+1}##. Find the interval of convergence and sum of the power series. Homework EquationsThe Attempt at a Solution According to the textbook: given the power series ##\sum a_n(x-c)^n## the radius of convergence...

Von neumann and bell pointed out that basically the non isomorphic fact that the spectrum $$\sigma(A+B)!=\sigma(A)+\sigma(B)$$ leads to contradictions. If we but replace the sum by $$A\otimes 1+1\otimes B$$ then the above inequality becomes an equality. This would make things much easier. We...

Homework Statement ##\sum\limits_{r=0}^n\frac{1}{^nC_r}=a##. Then find the value of $$\sum\sum\limits_{0\le i<j\le n}(\frac{i}{^nC_i}+\frac{j}{^nC_j})$$ Homework Equations I have used two equations which I derived myself. This is the first one. The second one is: 3. The Attempt at a...

Homework Statement Suppose that f:R->R satisfies the inequality ##|\sum\limits_{k=1}^n3^k[f(x+ky)-f(x-ky)]|<=1## for every positive integer k, for all real x, y. Prove that f is a constant function. Homework Equations None The Attempt at a Solution I tried taking f(x)=sinx and then using...

Good evening dearest physicians and mathematicians, I recently came across the so-called "Svein-Graham sum", and i wondered: is it possible to find a simple formula for evaluating it? \sum_{i=0}^k x\uparrow\uparrow i = \left .1+x+x^x+x^{x^x}+ ... +x^{x^{x^{x^{.^{.^{.^x}}}}}}\right \}k

Can I always say without reservation that for any two integral operators $K$ and $L$ defined as follows say $(Ky)(x)=\int_{a}^{b} \,k(x,s)y(s)ds$ that $||L||+||K-L||\ge||K||$ thanks Sarrah

Homework Statement If ##|P(x)|<=|e^{x-1}-1|## for all x> 0 where ##P(x)=\sum\limits_{r=0}^na_rx^r## then prove that ##|\sum\limits_{r=0}^nra_r|<=1## Homework Equations None The Attempt at a Solution ##P(1)=a_0+a_1+...## If the constants are positive, then ##P(1)<=|e^0-1|## So P(1)<=0 so...

Homework Statement [/B] Hello, thank you in advance for your help. I am calculating a Riemann sum with right hand endpoints. I hit a small snag, and I appreciate your help in getting me straight.Homework Equations f(x) = x2+ 1, over the interval [0,1]. This is problem number such-and-such from...

Find the sum of $$\sum_{n=1}^{\infty}\frac{1}{n2^{n}}$$ I tried manipulating it to match one of the Important Maclaurin Series and estimate the sum in that fashion but I cannot see to get it to match any. I was thinking of using $$\sum_{n=1}^{\infty}\frac{\left (\frac{1}{2} \right )^{n}}{n}$$...

Homework Statement How does a geometric series have a sum, or converge? Homework Equations Sum of Geometric Series = ##\frac {a} {1-r}## If r ≥ ±1, the series diverges. If -1 < r < 1, the series converges. The Attempt at a Solution How exactly does a infinite geometric series have a sum...

Homework Statement Find ## \sum_1^{23} tan^{-1}(\frac{1}{n^2+n+1}) ## Homework Equations ## tan^{-1}x + tan^{-1}y = tan^{-1}(\frac{x+y}{1-xy} )##The Attempt at a Solution I think we have to split the question in a form of relevant equation given above. First thing what should I do?