Sum Definition and 1000 Threads

Problem Statement : Let me copy and paste the problem as it appears in the text : Attempt : I haven't been able to make any significant attempt at solving this problem, am afraid. I tried to reduce all the higher submultiple angles ##2\theta, 4\theta, 8\theta## into ##\theta##, but the...

Hello, I've been trying to solve this problem for a while, and I found a technical solution which is too computationally intensive for large numbers, I am trying to solve the problem using Combinatorics instead. Given a set of integers 1, 2, 3, ..., 50 for example, where R=50 is the maximum...

I went ahead and tried to prove by induction but I got stuck at the base case for ## N =1 ## ( in my course we don't define ## 0 ## as natural so that's why I started from ## N = 1 ## ) which gives ## \sum_{k=0}^1 z_k = 1 + z = 1+ a + ib ## . I need to show that this is equal to ## \frac{1-...

Hey! 😊 A polynomial can be represented by a dictionary by setting the powers as keys and the coefficients as values. For example $x^12+4x^5-7x^2-1$ can be represented by the dictionary as $\{0 : -1, 2 : -7, 5 : 4, 12 : 1\}$. Write a function in Python that has as arguments two polynomials in...

I computed the double sum $$\sum_{i=0}^n\sum_{j=i+1}^n j = \sum_{i=0}^n\big(\frac{n(n+1)}{2}-\frac{i(i+1)}{2}\big)=\frac{n(n+1)(2n+1)}{6}$$ and realized the double sum is equal to $$\sum_{i=1}^ni^2$$ which leads to $$\sum_{i=0}^n\sum_{j=i+1}^n j = \sum_{i=1}^ni^2$$ Is there a proof of this...

Why is the total energy energy equal to the sum of translational kinetic energy and rotational kinetic energy? I understand the derivation KE = 1/2 I w^2 for a rigid object rotating around an axis: sum 0.5 * m_n * (v_T)^2 = sum 0.5 * m_n * (wr_n)^2 = 0.5 * w^2 * sum m_n r_n^2 = 0.5 * I * w^2...

AIUI, an algebraic is defined as a number that can be the solution (root) of some integer polynomial, and is any number that can be constructed via any binary arithmetic operation or unary root operation with arguments that are themselves algebraic numbers. I have been able to prove this for...

##\sum\limits_{k=0}^{n-m} \frac{\binom{n-m}{k}}{\binom{n}{k}}\frac{m}{n-k}=1##. Can be derived from question. For ##n\ge m##, pick ##m## marbled out of a set size ##n## labeled from ##1## to ##n##, what is probability distribution of minimum of the number labels on the marbles? The terms is...

Hi I was working on a physics problem and it was almost solved. Only the part that is mostly mathematical remains, and no matter how hard I tried, I could not solve it. I hope you can help me. This is the equation I came up with and I wanted to prove it: $$\lim_{n \rightarrow+ \infty} {...

I was supposed to find the relation among the coefficients $T$ and $R$ which represent the amplitude of the reflected electric field and the transmitted electric field respectively, that is, $$E_{R} = E_{i} R, E_{T} = E_{i} T$$ as well as the coefficients $t$ and $r$, that represent the...

Hi, I was attempting the following question and would appreciate any insight on how others would approach this game theory/probability-type question. Question: You have been chosen to play a game involving a 6-sided die. You get to roll the die once, see the result, and then may choose to...

In the homework I am asked to proof this, the hint says that I can use the triangle inequality. I was thinking that if both series go to a real number, a real number is just any number on the real number line, but how do I go from there,

1) If I take as definition of ##a_{lm}## following a normal distribution with mean equal to zero and ##C_\ell=\langle a_{lm}^2 \rangle=\text{Var}(a_{lm})##, and if I have a sum of ##\chi^2##, can I write the 2 lines below (We use ##\stackrel{d}{=}## to denote equality in distribution)...

Summary:: When we have only three classes (Orange, Banana and Other) and three features (Long, Sweet and Yellow), why P(Other|Long, Sweet, Yellow) + P(Banana|Long, Sweet, Yellow) is not equal to 1 when P(Orange|Long, Sweet, Yellow) = 0 ? In this example...

Let ##A## be a matrix of size ##(n,n)##. Denote the entry in the i-th row and the j-th column of ##A## by ##a_{ij}##, for some ##i,j\in\mathbb{N}##. For brevity, we call ##a_{ij}## entry ##(i,j)## of ##A##. Define the matrix ##X## to be of size ##(n,n)##, and denote entry ##(i,j)## of ##X## as...

This alternating series indentity with ascending and descending reciprocal factorials has me stumped. \frac{1}{k! \, n!} + \frac{-1}{(k+1)! \, (n-1)!} + \frac{1}{(k+2)! \, (n-2)!} \cdots \frac{(-1)^n}{(k+n)! \, (0)!} = \frac{1}{(k-1)! \, n! \, (k+n)} Or more compactly, \sum_{r=0}^{n} (...

Considering the below equality (or equivalency), could someone please explain how the bounds and indices are shifted? $$\sum_{i=2}^{k}(h_i/f_{i-1})=\int_{1}^{k}(h(i)/f(i))di$$

i do not seem to understand part ##ii## of this problem...mathematical induction proofs is one area in maths that has always boggled me :oldlaugh: let ##n=3, p=7, ⇒m=4## therefore, ##7=(4-3)(4+3)## ##7=1⋅7## ##1, 7## are integers...##p## is prime. i am attempting part ##iii## in a moment...

Given two subspaces ##U_1, U_2##, I understand the concept of direct sum $$ W= U_1 \oplus U_2 \iff W= U_1 + U_2, \quad U_1 \cap U_2 = \{ 0 \}$$ Where ##W## is a subspace of ##V##. I am trying to generalize it for more than ##2## subspaces, say ##3##. I thought of the following. $$ W= U_1...

##| V_1 \rangle \in \mathbb{V}^{n_1}_1## and there is an orthonormal basis in ##\mathbb{V}^{n_1}_1##: ##|u_1\rangle, |u_2\rangle ... |u_{n_1}\rangle## ##| V_2 \rangle \in \mathbb{V}^{n_2}_2## and there is an orthonormal basis in ##\mathbb{V}^{n_2}_2##: ##|w_1\rangle, |w_2\rangle ...

We know that Dirac Delta is not a function. However, I just talk about the numerical version of it that we use every day. We can simply represent the Dirac delta function as a limiting case of Gaussian distribution when the width of the distribution ##\sigma->0##. $$ \delta(x - \mu) =...

Let's say I want to study subalgebras of the indefinite orthogonal algebra ##\mathfrak{o}(m,n)## (corresponding to the group ##O(m,n)##, with ##m## and ##n## being some positive integers), and am told that it can be decomposed into the direct sum $$\mathfrak{o}(m,n) = \mathfrak{o}(m-x,n-x)...

This is a question that I saw in a textbook: "If the magnitude of a+b equals the magnitude of a+c then this implies that the magnitudes of b and c are equal. Is this true or false?" The textbook says that this statement is true, but I'm inclined to believe it is false. I made a quick sketch to...

Assume that $x_1,\,x_2,\,\cdots,\,x_n\ge -1$ and $\displaystyle \sum_{i=1}^n x_i^3=0$. Prove that $\displaystyle \sum_{i=1}^n x_i\le \dfrac{n}{3}$.

What I mean is the way that a product of cosines in which the angles increment the same amount is equal, with some extra terms, of the sum of the cosines. It is discussed here...

Two fair dice are rolled. What is the probability of rolling a sum that exceeds 3?

(a) I imagine there are several rectangles to represent the area under graph of p vs t then I try to make equation for the total area. Since the question asks about time when the container holds 22 fewer liters than it does at time t = 9, I think the total area of rectangles starting from t = b...

From @fresh_42's Insight https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/ Please discuss! We all live on a globe, a giant ball. The angles of a triangle on this ball add up to a number greater than ##180°##. And the amount by which the sum extends...

I want to evaluate the sum using Python and SymPy, I'm relatively new using SymPy and the first thing I tried was from sympy import exp, oo, summation, symbols n, x = symbols('n,x') summation(exp(-n*x), (n, 1, oo)) But it doesn't work, it just returns the unevaluated sum. I can make it...

$x,\,y$ and $z$ are non-negative real numbers that satisfy the following system: $x^2+y^2+xy=3\\y^2+z^2+yz=4\\z^2+x^2+xz=1$ Evaluate $x+y+z$.

(I know how to prove it). Prove that a finite sum of of independent normal random variables is normal. I suspect that independence may not be necessary.

I am reading the textbook Magnetism and Magnetic Materials by Coey and I am confused about how they grouped the terms and how they ended up getting the sums of L and S. My confusion lies in the two red boxes. Also, how is D even considered here when we have up to $2p_1$? And why would the spin...

I managed to write $$F_{\alpha\beta}F^{\alpha\gamma}=F_{0\beta}F^{0\gamma}+F_{i\beta}F^{i\gamma}$$ where $$i=1,2,3$$ and $$\gamma=0,1,2,3=\beta$$. How do I proceed?

How many different arithmetic sequences have the sum of the first n terms n^2? solution an= 2n-1.Does that mean there is only one arithmetic sequence?

For a pair of positive integers $(a,\,b)$, $Q(a,\,b)$ is defined by $Q(a,\,b)=\dfrac{a^2b+2ab^2-5}{ab+1}$. Let $(a_1,\,b_1),\,(a_2,\,b_2),\,\cdots, (a_n,\,b_n)$ be all pairs of positive integers such that $Q(a,\,b)$ is an integer. Calculate $\displaystyle \sum_{i=1}^n a_i$,

In an acute triangle $ABC$, prove that $\sqrt{\tan A}+\sqrt{\tan B}+\sqrt{\tan C} \ge \sqrt{\cot \left(\dfrac{A}{2}\right)}+\sqrt{\cot \left(\dfrac{B}{2}\right)}+\sqrt{\cot \left(\dfrac{C}{2}\right)}$.

Consider polarized light crossing a sharp boundary between two volumes, each of a different but uniform refraction index ##n_1## or ##n_2##. Prove that the sum of the transmission and reflection coefficients of this light ##R+T=1##, where $$R \equiv {I_R \over I_I} = \left( {E_{0_R} \over...

Good day I really don't understand how they got this result? for me the sum of the Fourier serie of of f is equal to f(2)=log(3) any help would be highly appreciated! thanks in advance!

Can someone please explain why the formula for midpoint approximation looks like the equation above instead of something like $$M_n=(f(\frac{x_0+x_1}2)+f(\frac{x_1+x_2}2)+\cdots+f(\frac{x_{n-1}+x_n}2))\frac{b-a}n$$? Thanks in advance!

Greeting I'm trying to study the convergence of this serie I started studying the absolute convergence because an≈n^(2/3) we know that Sn will be divergente S=∝ so arcatn (Sn)≤π/2 and the denominator would be a positive number less than π/2, and because an≈n^(2/3) and we know 1/n^(2/3) >...

Hawking and Hartle proposed a well-known model which postulated a sum over all possible histories considering all compact euclidean metrics to explain the origin of the universe (this is called the No Boundary model). I was wondering whether there is any model or theory (related to cosmology)...

Let us suppose we have an equation such that $$N = \sum_{i=1}^N ix_i = x_1 + 2x_2 + 3x_3 + ...+Nx_N$$ and we also know that the solutions (i.e ##x_i##) ranges from ##\{0, N\}##. For example, if ##N=4## we would have $$x_1 + 2x_2 + 3x_3 + 4x_4 = 4$$ and ##x_1,x_2,x_3,x_4## will range from...

For any natural number $n$, let $S(n)$ denote the sum of the digits of $n$. Find the number of all 3-digit numbers $n$ such that $S(S(n))=2$.

Let ##n=\dim X## and ##m=\dim Y##. Define a basis for ##X: y_1,...,y_m,z_{m+1},...,z_n##. The first ##m## terms are a basis for ##Y##. The remaining ##n-m## terms are a basis for its complement w.r.t ##X##. Let's call it ##Z##. ##X## is the direct sum of ##Y## and ##Z##; denote it as ##X=Y+Z##...

What does Feynman's sum over histories mean to the interpretation of our world? Does it mean that we (or a particle) do not have a definite history, but only the most probable one?

Hey! :giggle: Let $N_j$, $j=-k,\ldots , m-1$ the normalized B-splines of the set of nodes $x_0, \ldots , x_m$ of degree $k$. Show that $$\sum_{j=-k}^{m-1}N_j(x)=1 \ \text{ for all } x\in [x_0, x_m]$$ A formula with divided differences is \begin{align*}&N_j(x)=(x_{j+k+1}-x_j)B_j(x) \\&...

The "egg" initially spun around axis 1 with at ##\omega_s##. After being disturbed, it has started to possesses angular velocities along 2 and 3. The question is to find the rotational speed of ##\vec \omega=\vec\omega_1+\vec\omega_2+\vec\omega_3## to a fixed observer. It is calculated that...

if x_{I}, I = {1,2,...,2019} is a root of P(x) = ##x^{2019} +2019x - 1## Find the value of ##\sum_{1}^{2019}\frac{1}{1-\frac{1}{X_{I}}}## I am really confused: This polynomial jut have one root, and this root is x such that 0 < x < 1, so that each terms in the polynomial is negative. But the...

$\tiny{gre.al.13}$ For which of the following conditions will the sum of integers m and n always be an odd integer.? a. m is an odd integer b. n is an odd integer c. m and n both are odd integers d. m and n both are even integers e. m is an odd integer and n is an even integerI chose e just...

Let $a,\,b,\,c,\,d,\,e,\,f$ be positive integers and $S=a+b+c+d+e+f$. Suppose that the number $S$ divides $abc+def$ and $ab+bc+ca-de-ef-df$, prove that $S$ is composite.