Sum Definition and 1000 Threads

I ran into some issues when trying to calculate the lower Riemann sum of $$f\left(x\right)={x}^{3}$$, $$x\in[0,1]$$ I am asked to use the standard partition $${P}_{n}$$ of $$[0,1]$$ with n equal subintervals and evaluate $$L(f,{P}_{n})$$ and $$U(f,{P}_{n})$$ What I did: $$L(f,{P}_{n}) =...

I would like to approximate a plane electromagnetic wave with a very large sum of the following. Let an infinite line, say the z axis, have a electric polarization on that line and perpendicular to that line, say the x direction to be specific given by, P(z,t) = pcos(kz-ωt). The polarization...

I think that we have to get all 2 digit odd numbers that can be expressed as the sum of 2 primes and subtract that from 45, so I think that the answer would be 45-(number of 2 digit integers n that are prime and have n-2 be prime as well)?

In integral calc, you add up very small areas to find the total area under the curve. So it would be f(x1)Δx + f(x2)Δx+ ..., summed up. But what if you wanted to find out the sum of all heights under the curve? So it would be something like f(x1) + f(x2) + ... I'm thinking the formulation would...

Show that the sum of the roots of the equation x^2 + px + q = 0 is -p. I need help with the set up. Is the discriminant involved here?

Homework Statement Two ladders, 4.00 m and 3.00 m long, are hinged at point A and tied together by a horizontal rope 0.90 m above the floor (Fig. P11.89). The ladders weigh 480 N and 360 N, respectively, and the center of gravity of each is at its center. Assume that the floor is freshly waxed...

Homework Statement Show that ##a \sin x + b\cos x = c \sin (x + \theta)##, where ##c = \sqrt{a^2 + b^2}## and ## \displaystyle \theta = \arctan (\frac{b}{a})## Homework EquationsThe Attempt at a Solution We see that ##c \sin (x + \theta) = c \cos \theta (\sin x) + x \sin \theta (\cos x)##. So...

$\textsf{Find the sum of the series}\\$ \begin{align*}\displaystyle S_{n}&=\sum_{n=1}^{\infty} \frac{4}{(4n-1)(4n+3)}=\color{red}{\frac{1}{3}} \\ \end{align*} $\textsf{expand rational expression } $ \begin{align*}\displaystyle \frac{4}{(4n-1)(4n+3)}...

I've been reading my physics book and there they derived the formula ∑τ = Iα where τ is torque, I is moment of inertia of a rigid body and α is the angular acceleration. They did by taking an arbitrary particle on the rigid body with an applied external force tangent to the rotation. τ1 = Ftan *...

Homework Statement What is the value of ## \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + ... ## ? Homework Equations [/B] I have no idea since it's neither a geometric nor arithmatic seriesThe Attempt at a Solution [/B] My Calculus purcell book tells me that it is e - 1 ≈...

Say you have two energy eigenstates ##\phi_1## and ##\phi_2##, corresponding to energies ##E_1## and ##E_2##. The particle has a 50% chance of having each energy. The wavefunction would thus be ##\psi=\frac{\phi_1}{\sqrt{2}}+\frac{\phi_2}{\sqrt{2}}## Even though the coefficients are normalized...

i want to know if any real function can be expressed as: f(x)=g(x)+h(x) such as g(x) is an increasing function and h(x) is a decreasing function? thanks

Homework Statement I need to find the pdf of sum of "n" iid non central chi-square distributed RV's. Homework Equations The PDF of the non-central chi-square RV is given herehttps://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution The Attempt at a Solution i tried to find the...

Homework Statement Consider a sequence of non negative integers x1,x2,x3,...xn which of the following cannot be true ? ##A)\sum ^{\infty }_{n=1} x_{n}= \infty \space and \space \sum ^{\infty }_{n=1} x_{n}^{2}= \infty## ##B)\sum ^{\infty }_{n=1} x_{n}= \infty \space and \space \sum ^{\infty...

The definite integral of a function ##f(x)## from ##a## to ##b## as the limit of a sum is: $$\int_a^bf(x)dx=\lim_{h\rightarrow 0}h(f(a)+f(a+h)+.. ..+f(a+(n-2)h)+f(a+(n-1)h))$$ where ##h=\frac{b-a}{n}##. So, replacing ##h## with ##\frac{b-a}{n}## gives: $$\lim_{n\rightarrow...

Homework Statement ##1729## can be written as ##12^3 + 1^3## and ##9^3 + 10^3## and ##7(10 + 9)(12 + 1)##. If ##x^3 + (7 - x)^3 = 1729##, use the above to find ##x##. ##x## is a non-integer Homework Equations ##1729 = 12^3 + 1^3 = 9^3 + 10^3 = 7(10 + 9)(12 + 1) = x^3 + (7 - x)^3## The Attempt...

Hey guys! I just have a question regarding finding the sum of an infinite series. Attached is the image of the question. I've tried to use the ratio test but it doesn't give me the result I need which happens to be 1/sqrt(2). I feel like this is one of those power series questions, but I'm not...

Let V be a vector space. If U 1 and U2 are subspaces of V s.t. U1+U2 = V and U1 and U1∩U2 = {0V}, then we say that V is the internal direct sum of U1 and U2. In this case we write V = U1⊕U2. Show that V is internal direct sum of U1 and U2if and only if every vector in V may be written uniquely...

Homework Statement Hi, I am trying to follow the working attached which is showing that the average energy is equal to the most probable energy, denoted by ##E*##, where ##E*## is given by the ##E=E*## such that: ##\frac{\partial}{\partial E} (\Omega (E) e^{-\beta E}) = 0 ## MY QUESTION...

Homework Statement Find all integer solutions to x2 + y2 + z2 = 51. Use "without loss of generality." Homework Equations The Attempt at a Solution My informal proof attempt: Let x, y, z be some integers such that x, y, z = (0 or 1 or 2 or 3) mod 4 Then x2, y2, y2 = (0 or 1) mod 4 So x2 +...

What is the sum of all the even integers between 1 and 101? Is there an easier way besides using the formula: (B-A+1)(B+A)/2? It just takes too much time.

(From Hoffman and Kunze, Linear Algebra: Chapter 6.7, Exercise 11.) Note that ##V_j^0## means the annihilator of the space ##V_j##. V* means the dual space of V. 1. Homework Statement Let V be a vector space, Let ##W_1 , \cdots , W_k## be subspaces of V, and let $$V_j = W_1 + \cdots + W_{j-1}...

I found a deduction to determinate de sum of the first n squares. However there is a part on it that i didn't understood. We use the next definition: (k+1)^3 - k^3 = 3k^2 + 3k +1, then we define k= 1, ... , n and then we sum... (n+1)^3 -1 = 3\sum_{k=0}^{n}k^{2} +3\sum_{k=0}^{n}k+ n The...

Hi. I try to solve the integral $$\int_{0}^{1} x^{x} dx$$ Through sums of riemann But I came to the conclusion that the result is 0 that is wrong $$\int_{0}^{1} x^{x} dx = \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{k=1}^{n} \left ( \frac{k}{n} \right )^{\frac{k}{n}}$$ $$= \lim_{n\rightarrow...

Hi there, I need help with the following situation. Apologies if I'm not using the correct arithmetic terms! Variables: d,e,f e + f = g d / g = j j x e = K j x f = L K + L = M d = M the above situation is a simplified problem, which is easily solvable. Here's where I run into trouble...

The problem I want to calculate $$\sum^n_{k=1} \frac{4}{1+ \left(\frac{k}{n} \right)^2} \cdot \frac{1}{n}$$ when ##n \rightarrow \infty## The attempt ## \sum^n_{k=1} \underbrace{f(\epsilon)}_{height} \underbrace{(x_k-x_{k-1})}_{width} \rightarrow \int^b_a f(x) \ dx ##, when ##n \rightarrow...

Evaluate $$\sum_{n>1} \frac{3n^2+1}{(n^3-n)^3}$$.

$a_1,a_2,...,a_{100}\in \begin{Bmatrix} 1,2,3,-----,100 \end{Bmatrix}$ $S=\dfrac{1}{\sqrt{a_1}}+\dfrac{1}{\sqrt{a_2}}+\cdots+\dfrac{1}{\sqrt{a_{100}}}=12.5$. Prove that at least two of the numbers are equal

Evaluate $$\tan^4 10^\circ+\tan^4 50^\circ+\tan^4 70^\circ$$ without the help of a calculator.

Homework Statement Find the sum of the following series: $$ \left( \frac 1 2 + \frac 1 4 \right) + \left( \frac 1 {2^2} + \frac 1 {4^2} \right) +~...~+ \left( \frac 1 {2^k} + \frac 1 {4^k} \right) +~...$$ Homework Equations $$ \sum_{n = 1}^{\infty} \left( u_k+v_k \right) = \sum_{n =...

The natural numbers $a_1,\,a_2,\,\cdots,\,a_{100}$ are such that $\dfrac{1}{\sqrt{a_1}}+\dfrac{1}{\sqrt{a_1}}+\cdots+\dfrac{1}{\sqrt{a_1}}=20$. Prove that at least two of the numbers are equal.

[Please excuse the screengrabs of the fomulae - I'll get around to learning TeX someday!] 1. Homework Statement Find the sum of this series (answer included - not the one I'm getting) The Attempt at a Solution So I'm trying to sum this series as a telescoping sum. I decomposed the fraction...

First term of the progression is 3 & the common difference is 4 Find the sum of the first 20 terms of the progression that is obtained by removing the terms in the even positions of the given progressions, such as the second term,fourh term, sixth term. Formula preferences For the sum of an...

$n\in N,\,\,and \,\, a_1,a_2,a_3,-------,a_n\in Z$ $if \,\, a_+a_2+a_3+-----+a_n=a_1\times a_2\times a_3\times------\times a_n=2006$ $find \,\, min(n)$

Consider a flat 2D rigid body rotating about an axis perpendicular to the body passing through a point P that is (1) in the same plane as the body and (2) different from the body's center of mass (CM). In this case does Theorem 7.1 (eqn 7.9) still apply? In the last step of the derivation of...

$\tiny{206.8.5.49}$ $\textsf{Express the integrand as sum of partial fractions}$ \begin{align} && I_{49}&=\int\frac{30s+30}{(s^2+1)(s-1)^3}\, ds& &(1)& \\ &\textsf{expand}& \\ && &=\displaystyle 15\int\frac{1}{(s^2+1)}\, ds -15\int\frac{1}{(s-1)^2}\, ds +30\int\frac{1}{(s-1)^3}\, ds&...

uhh, how would we get a better way?

Homework Statement Given ##v = {\left\{ {v}_{i} \right\}}_{i = 1}^{\infty}## and defining ## {v}_{n}^{\left( k \right)} = {v}_{n - k} ## (Shifting Operator). Prove that there exist ## \alpha > 0 ## such that $$ \sum_{k = - \infty}^{\infty} {2}^{- \left| k \right|} \left \langle {v}^{\left (...

Can x/(z+y) be written as an infinite sum?

Find the minimum of the sum: \[\sum_{i=1}^{5}x_i\], where $x_i \ge 0$ and $\sum_{i<j}|x_i-x_j| = 1.$

Homework Statement Find the sum of the given infinite series. $$S = {1\over 1\times 3} + {2\over 1\times 3\times 5}+{3\over 1\times 3\times 5\times 7} \cdots $$ 2. Homework Equations The Attempt at a Solution I try to reduce the denominator to closed form by converting it to a factorial...

$\tiny{206.10.5.84}$ \begin{align*} \displaystyle S_{84}&=\sum_{k=1}^{\infty} \frac{(4x)^k}{5k}\\ \end{align*} $\textsf{ ratio test}$ $$\frac{a_{n+1}}{a_n} =\frac{ \frac{(4x)^{k+1}}{5(k+1)}}{ \frac{(4x)^k}{5k}} =\frac{4xk}{k+1} $$ $\textsf{W|A says this converges at $4|x|<1 $ so how??}$

$\tiny{206.b.46}$ \begin{align*} \displaystyle S_{book}&=\sum_{k=1}^{\infty} \frac{8^k}{k! }=0\\ S_{TI}&=\sum_{k=1}^{\infty} \frac{8^k}{k! }=e^8-1\\ \end{align*} $\textsf{ 2 different answers?}$

What's the reason which implies that we can't have a formula for the sum of HP. https://en.m.wikipedia.org/wiki/Harmonic_progression_(mathematics) Wikipedia gave a reson , can you elaborate it.

$\tiny{206.10.3.17}$ $\textsf{Evaluate the following geometric sum.}$ $$\displaystyle S_n=\frac{1}{2}+ \frac{1}{8}+\frac{1}{32}+\frac{1}{128}+\cdots + \frac{1}{8192}$$ $\textsf{This becomes}$ $$\displaystyle S_n=\sum_{n=1}^{\infty}\frac{1}{2^{2n-1}}=\frac{2}{3}$$ $\textsf{How is this morphed...

$\tiny{242.10.3.27}$ evaluate $$S_j=\sum_{j=1}^{\infty}3^{-3j}=$$ rewrite $$S_j=\sum_{j=1}^{\infty} 27^{j-1}$$ using the geometric formula $$\sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{1-r}, \left| r \right|<1$$ how do we get $a$ and $r$ to get the answer of $\frac{1}{26}$ ☕

Homework Statement 4 dice are rolled. Find probability that sum is 20. Homework Equations If a dice is rolled the outcome can be 1, 2, 3, 4, 5, 6 The Attempt at a Solution Well the combinations for sum to be 20 are: 5, 5, 5, 5 = 20 5, 5, 6, 4 = 20 6, 6, 5, 3 = 20 6, 6, 4, 4 = 20 6, 6, 6...

Problem: Let $k$ be a natural number, and $k+1 \equiv 0 \:\: (mod\:\:24)$ Show, that the sum of $k$´s divisors is also divisible by $24$. Solution: First, note that since $k = 4n_1+3$ for some $n_1\in \mathbb{N}$, $\sqrt{k}$ is not a natural number. Let $p_1,p_2,…,p_m < \sqrt{k}$ be all...

I've lately began working with Newtons laws problems at school again, and I've already ran into a few problems. When making calculations and solving problems, it is often nessecary to understand when forces are equal to zero, and when they are not. Since every force has an equal and opposite...

Homework Statement I've been given the spherical harmonics ##Y_{l,m}## for the orbital quantum number ##l=1##. Then told to calcute the sum of their squares over all values of m and explain the significance of the result. Homework Equations ##Y_{1,1} =...