Sum Definition and 1000 Threads

$\tiny{206.b.46}$ \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} =...

Theorem: Let ##A_1, A_2, ..., A_k## be finite, disjunct sets. Then ##|A_1 \cup A_2 \cup \dots \cup A_k| = |A_1| + |A_2| + \dots + |A_k|## I will give the proof my book provides, I don't understand several parts of it. Proof: We have bijections ##f_i: [n_i] \rightarrow A_i## for ##i \in [k]##...

Hello everyone, I have an issue solving the following problem: You're on a mathematical Olympiad, there are m medals and it lasts for n days. First day committee gives U_{1}=1+\frac{1}{7}(m-1) medals. On the second day U_{2}=2+\frac{1}{7}(m-2-U_{1}) medals, and so on... On the last day...

Homework Statement The average value of N measurements of a quantity ##v_i## is defined as $$ \langle v \rangle \equiv \frac {1}{N} \sum_{i=1}^Nv_i = \frac {1}{N}(v_1 + v_2 + \cdots v_N)$$ The deviation of any given measurement ##v_i## from the average is of course ##(v_i - \langle v...

<<Moderator's note: moved from a technical forum, so homework template missing.>> I found a problem in Boas 3rd ed that asks the reader to use S_n = 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + ... to show that the harmonic series diverges. They specifically want this done using the test...

Here is the given problem... And I first approached it by drawing the xy footprint to get my theta and radius limits to convert to polar. Then I overlooked the original xy function and pretty much took the area of that footprint (highlighted in green.) That gave me a very nice number...

Homework Statement Show that any sum: Asin(α) + Bcos(α) can be written as : C sin(α+ϕ) 2. Homework Equations The Attempt at a Solution i can express cos(a) as as sin(90-a), and then try to use the formula that adds sines, but it gives the form of cos*sin. [/B]

prove by induction for all positive integers n: 1+5+9+13+...+(4n-3)= n/2(4n-2) i tried this by trying to prove n/2(4n-2)+ (4(k+1)-3) = k+1/2(4(k+1)-2) but it did not work out for me.

I am looking for a way to simplify the following expression: ##\sum\limits_{n=1}^{N}\ \sum\limits_{k=0}^{N-1}\ \sum\limits_{k'=0}^{N-1}\ \tilde{p}_{k}\ \tilde{p}_{k'}\ e^{2\pi in(k+k')/N}##. I presume that the sum of the exponentials over ##n## somehow reduce to a Kronecker delta. Am I wrong?

Homework Statement Prove that, $$\sum _{n=1,3,5...} \frac{1}{n} e^{-nx} \sin{ny} = \frac{1}{2}\tan^{-1} (\frac{\sin{y}}{\sinh{x}})$$ Homework Equations $$\tan^{-1}{x} = x - \frac{x^3}{3} +\frac{x^5}{5} - ... $$ 3. The Attempt at a Solution $$\sum _{n=1,3,5...} \frac{1}{n} e^{-nx}...

Homework Statement \sin (x) = \frac{2}{3} and \sec (y) = \frac{5}{4}, where x and y lie between 0 and \frac{\pi}{2} evaluate \sin (x + y) Homework Equations Looked over some trig laws, don't think I saw anything that's too relevant. There \sec (x) = \frac{1}{\sin (x)} The Attempt at a...

My first thought was simply that the Fourier transform of a sum of Gaussians functions that are displaced from the origin by different amounts would just be another sum of Gaussians: F{G1(x) + G2(x)} = F{G1(x)} + F{G1(x)} where a generalized shifted Gaussian is: G(x) = G0exp[-(x - x0)2 / 2σ2]...

Homework Statement Homework Equations Summation The Attempt at a Solution I know I could have simplified (3n-2)^3 +(3n-1)^3 -(3n)^3 and put the formulas in but I wonder is there any other method (I was thinking about grouping the terms, but to no avail) to work this out.

Calculate the sum for the infinite geometric series $4+2+1+\frac{1}{2}+...$ all I know is the ratio is $\frac{1}{2}$ $\displaystyle\sum_{n}^{\infty}a{r}^{n}$ assume this is used

Let \[f(x) = \frac{a^{2x}}{a^{2x}+a}, \;\;\; a \in \Bbb{N}.\]Find the sum:\[ \sum_{j=0}^{2016}f \left ( \frac{j}{2016} \right )\]

Homework Statement I was given a problem with a list of sums of sinusoidal signals, such as Example that I made up: x(t)=cos(t)+5sin(5*t). The problem asks if a given expression could be a Fourier expansion. Homework Equations [/B]The Attempt at a Solution My guess is that it has something to...

Homework Statement Let f [a, b] → R be a non-decreasing function; that is, f(x1) ≤ f(x2) for any x1, x2 ∈ [a, b] with x1 ≤ x2. So f attains a minimum value of m = f(a) and a maximum value of M = f(b) on [a, b]. Let Pn be a regular partition of [a, b] into n subintervals, each of length ∆x = (b...

So I am working on this simple proof, but am confused about the term "external angle." The problem says that if ##a##, ##b##, and ##c## are external angles to a triangle, then ##a + b + c = 360##. However, is seems that the vertex of each triangle has two possible external angles, since there...

Hello all, Suppose I have the following summation ##X=\sum_{k=1}^KX_k## where the ##\{X_k\}## are independent and identically distributed random variables with CDF and PDF of ##F_{X_k}(x)## and ##f_{X_k}(x)##, respectively. How can I find the CDF of ##X##? Thanks in advance

Any idea where I'm going wrong here? It's failing some test cases. I thought my solution was straightforward (if not brute force). using System; using System.Collections.Generic; using System.IO; using System.Linq; class Solution { static void Main(String[] args) { int k =...

This has turned out to be a long question to type out so I apologise, but I don't think it's too hard to follow or read through quickly and I believe the actual question itself may not be too complicated once I get round to asking it. You can possibly skip to the last few paragraphs and still be...

Hello. The quantum mechanics textbook shows the relation of J1 + J2 + J3 ≠ J1 + (J2 + J3). I believe Ji is total angular momentum operator for ith group of electrons (but actually I have not seen J1 operator while I have seen J12 operator so far). I don't know how to prove J1 + J2 + J3 ≠ J1 +...

I'm trying to find pythagorean triples that sum to 60. Just from memory, I kow that 3-4-5 and 5-12-13, scaled to some factor, will give triples that sum to 60. These seem to be the only ones that sum to 60, but how can I be sure that there aren't more triples that sum to 60?

Prove \frac{10}{\sqrt{11^{11}}}+\frac{11}{\sqrt{12^{12}}}+\cdots+\frac{2015}{\sqrt{2016^{2016}}}\gt \frac{1}{10!}-\frac{1}{2016!}

Ok so when differentiating 1/(1-x)= Σ xn from n=0 to infinity the book says it is 1/(1-x)^2 = Σ n*(x)n-1 from n=1 to infinity i don't understand why the original sum starts at 0 and then the derived sum starts at 1

Find the sum of this series: $$ \sum_{n=1}^\infty \frac{n}{(n+1)!} $$ I'm really struggling with this one.. Any help will be highly appreciated. Thanks you.

Homework Statement If the nth partial sum of a series ##\sum_{n=1} ^\infty a_{n}## is ##S_{n} = \frac {n-1} {n+1}## Find ##a_{n}## and ##\sum_{n=1}^\infty a_n## Homework Equations ##S_{n} - S_{n-1}= a_{n}## ##\lim_{n \rightarrow +\infty} {S_{n}} = \sum_{n=1}^\infty a_n = S## The Attempt at a...

The problem I want to calculate the following sum $$ \sum^{5}_{k=2} \frac{k(-1)^k}{2^k} $$ The attempt I wrote ## \frac{(-1)^k}{2^k} ## as ##\frac{1}{(-2)^k}##. I was hoping that I could calculate the sum ## \sum^{5}_{k=2} \frac{k(-1)^k}{2^k} ## by multiplying the sums ##\sum^{5}_{k=2} k##...

Let $a,\,b$ and $c$ be positive real numbers for which $a+ b + c = 1$. Prove that \frac{a^3}{b^2+c^2}+\frac{b^3}{c^2+a^2}+\frac{c^3}{a^2+b^2}\ge \frac{1}{2}.

I have a model in which, for each store, predicted revenues are perturbed by a multiplicative shock: R = e^\eta r where r is predicted and R is observed. \eta is mean zero. I can find \eta as follows: \ln( r) - \ln( R) = \eta . I'm summing the squares of the \eta's. However, there are...

Prof Adams does something rather strange, starting from 14:35 minutes in this lecture -- http://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/lecture-videos/lecture-9/ He reminds us that for complex scalars, ##c^2+d^2=(c-id)(c+id)## and then proceeds to do the same with...

Hello! I am reading something about applications of group theory in quantum mechanics and I got confused about the difference between direct sum and direct product. In many places I found that they mean the same thing. However, the ways I found them defined in the book I read from, seem to be...

Homework Statement What is the sum of the angles of a spherical triangle formed on the surface of a sphere of radius R? The triangle is formed by the intersections of the arcs of great circles. Let A be the area of the surface of the sphere enclosed by the triangle. This question is a...

Homework Statement Let A,B be square matrices of order n. n>=2 lets A and B be matrices of Rank 1. What are the options of the Rank of A+B ? Homework EquationsThe Attempt at a Solution I know that there are 3 possibilities, 2, 1 , 0. Just having trouble with coming up with a formula. i tried...

I was recently researching into some string theory when i came across the following summation: The sum of all natural numbers is -1/12, now I'm still wrapping my head around the context of the application within critical string dimensions, but is this summation valid? And if not, why it being...

Find the sum for the series $$\frac{5}{3}+2+\frac{12}{5}+...$$ This equals $$\frac{25}{15}+\frac{30}{15}+\frac{36}{15}+...$$ So the numerator increases by 4+k from the previous numerator But unable to set up $$\sum_{k+1}^{\infty}f(x)$$ The series should go to $\infty$ since the terms only...

nmh{2000} index{expanded form} write each expression in expanded form and then find the sum $ \begin{array}{l}...

Find the sum of the first $17$ terms of the arithmetic series: $8+\sqrt{7}$, $6$, $4-\sqrt{7 }$... $a_1=8+\sqrt{7}$; $n=17$; $d=2+\sqrt{7 }$ $\displaystyle\sum_{k=1}^{n}(a_1-kd)=136 \sqrt{7 }-170$ Don't have book answer for this? Much Mahalo

I have a Stats exam on Wednesday and while I thought I was quite well-versed, I've gone back over to the very basics only to find myself confused at what should be introductory. Suppose I have a continuous random variable modeled by a probability density function: $$f(x)=2x$$ Obviously the...

I'm reading through my quantum physics lecture notes (see page 216 of the lecture notes for more details) and under the ladder operators section there is a discussion of the expectation value of ##L_x## for a state ##\psi = R(r) \left( \sqrt{ \frac{2}{3}} Y_{11} - \sqrt{ \frac{1}{3}} Y_{10}...

Suppose if I applied a 4N force and then 2N force on an object , what will be total force. Note I didn't said simoultaneously, I mean one after the other, then what will be the total force , I think 6N , i know about vector sum and etc.. but I think this question doesn't makes any sense...

Hello I am reading "The Theory of Numbers, by Robert D. Carmichael" and stuck in an exercise problem, Find numbers x such that the sum of the divisors of x is a perfect square. I know sum of divisors of a x = p_1^{{\alpha}_1}.p_2^{{\alpha}_1}...p_n^{{\alpha}_1} is Sum of divisors...

This is probably a very trivial question, but my brain isn't "playing ball" today so I'm hoping someone can help me with this. Suppose I have a system of ##N## mutually interacting particles, then the force on the ##i##-th particle due to the other ##N-1## particles is given by...