Sum Definition and 1000 Threads

Evaluation of \displaystyle \sum_{r=1}^\infty \left(\frac{1}{36r^2-1}+\frac{2}{(36r^2-1)^2}\right)

Homework Statement >Find the sum of the roots, real and non-real, of the equation x^{2001}+\left(\frac 12-x\right)^{2001}=0, given that there are no multiple roots. While trying to solve the above problem (AIME 2001, Problem 3), I came across three solutions on...

##\displaystyle \sum_{n=1}^\infty\frac{1}{n^2+n/2}## converges by the direct comparison test: ##\displaystyle \left|\frac{1}{n^2+n/2}\right| \le \left|\frac{1}{n^2}\right|##, and ##\displaystyle \sum_{n=1}^\infty\frac{1}{n^2} = \frac{\pi^2}{6}##. But what if we want to show that ##\displaystyle...

For 0 < x< 1, find the sum: \[\frac{2x-1}{1-x+x^2}+\frac{4x^3-2x}{1-x^2+x^4}+\frac{8x^7-4x^3}{1-x^4+x^8}+ ...\]

<Moderator's note: Moved from a technical forum and thus no template.> So, I have this problem and I am stuck on a sum. The problem I was given is the following: The probability of a given number n of events (0 ≤ n < ∞) in a counting experiment per time (e.g. radioactive decay events per...

Homework Statement (x.y)ER+ that means x and y >=0 Homework Equations Prove that n√(x+y)<=n√x + n√y The Attempt at a Solution

Why are the resultant's X and Y components of two forces equal the sum of the X and Y components of the two forces?

Homework Statement Let ##V = \mathbb{R}^4##. Consider the following subspaces: ##V_1 = \{(x,y,z,t)\ : x = y = z\}, V_2=[(2,1,1,1)], V_3 =[(2,2,1,1)]## And let ##V = M_n(\mathbb{k})##. Consider the following subspaces: ##V_1 = \{(a_{ij}) \in V : a_{ij} = 0,\forall i < j\}## ##V_2 =...

Homework Statement For each ##n\in\mathbb{N}##, let the finite sequence ##\{b_{n,m}\}_{m=1}^n\subset(0,\infty)## be given. Assume, for each ##n\in\mathbb{N}##, that ##b_{n,1}+b_{n,2}+\cdots+b_{n,n}=1##. Show that ##\lim_{n\to\infty}( b_{n,1}\cdot a_1+b_{n,2}\cdot a_2+\cdots+b_{n,n}\cdot a_n) =...

Hi All Been investigating lately ways to sum ordinarily divergent series. Looked into Cesaro and Abel summation, but since if a series is Abel Mable it is also Cesaro sumable, but no, conversely,haven't worried about Cesaro Summation. Noticed Abel summation is really a regularization...

Homework Statement The distance between the centres of the Earth and the moon is 60 times the radius of the earth. Calculate the centripetal acceleration of the moon. Acceleration due to gravity on the Earth's surface is 10m/s. Homework Equations Centripetal acceleration= v^2/R Orbital...

There is a problem in a PreCalculus book that I'm going over that states: Express the sum ##\frac{1}{2⋅3}+\frac{1}{3⋅4}+\frac{1}{4⋅5}+...+\frac{1}{2019⋅2020}## as a fraction of whole numbers in lowest terms. It goes on to state that each term in the sum is of the form...

I just began graduate school and was struggling a bit with some basic notions, so if you could give me some suggestions or point me in the right direction, I would really appreciate it. 1. Homework Statement Given an infinite base of orthonormal states in the Hilbert space...

Homework Statement Suppose that ## \mathbb {V}_1^{n_1} ## and ## \mathbb {V}_2^{n_2} ## are two subspaces such that any element of ## \mathbb {V}_1^{n_1} ## is orthogonal to any element of ## \mathbb {V}_2^{n_2} ## . Show that dimensionality of ## \mathbb {V}_1^{n_1} + \mathbb {V}_2^{n_2}...

Kate, Nora, and Devi shared a sum of money. Kate received 24 dollars and Nora received x dollars more than Kate. Devi received 2x dollars more than kate a) Find the sum of money shared in terms of x. my answer: total = 24 + (x+24) + 2(24)b) Nora received $30. Find the total sum of money shared...

THE QUESTION By using Riemann’s sum, synthesise a mathematical model for finding the exact volume of any ‘tepee’ tent of side s and height h. HERE'S WHAT I HAVE Am currently stuck on writing a side length for the hexagon at any height 'x'

Let $a,\,b$ and $c$ be real numbers such that $a+b+c=ab+bc+ac=-\dfrac{1}{2}\\abc=\dfrac{1}{8}$ Evaluate $a^{\tiny\dfrac{1}{3}}+b^{\tiny\dfrac{1}{3}}+c^{\tiny\dfrac{1}{3}}$.

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This is what i have so far We can find the exact volume of any shape using: V= $$int[a,b] A(x) dx$$ Where,A(x)is the cross-sectional area at height x and [a,b] is the height interval We know that the horizontal cross-sections are hexagonal $$∴A=(3√3)/2 a^2$$ Where a,is the length of a side...

Q: The sum of the present ages of Vatha and Chris is 36. In 4 years time, the sum of their ages will equal twice Vatha's present age. How old are they now?A: Vatha:22, Chris: 14 (from the back of the textbook, only I'm not sure how to get here)I'm a little stuck with this one. If you could...

The partition function should essentially be the sum of probabilities of being in various states, I believe. Why is it then the sum of Boltzmann factors even for fermions and bosons? I've never seen a good motivation for this in literature.

##X_i## is an independent and identically distributed random variable drawn from a non-negative discrete distribution with known mean ##0 < \mu < 1## and finite variance. No probability is assigned to ##\infty##. Now, given ##1<M##, a sequence ##\{X_i\}## for ##i\in1...n## is said to meet...

Homework Statement I am trying to understand the very last equality for (let me replace the tilda with a hat ) ##\hat{P_{X}(K)}=\hat{P(k_1=k_2=...=k_{N}=k)}##(1) Homework Equations I also thought that the following imaginary exponential delta identity may be useful, due to the equality of...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.10 ... ... Proposition 4.2.10 reads as follows:My questions are as follows:Question 1 In the...

I have the following series that I came up with in doing a problem: ##\displaystyle \sum_{n=0}^{\infty} \frac{1}{2^{n+1}(n+1)}##. I looked at WolframAlpha and it says that this series converges to ##\log (2)##. Is it possible to figure this out analytically?

Let's suppose there are 4 types of coin, 1p 2p 5p and 10p. The problem is, we need to find total combinations which the sum of the coins gives us the 200p. I am trying to find a mathematical equation to solve the problem but I am stuck. First I started to think algebraically. And I have this...

⇒Homework Statement [/B] Calculate ##S + T## and determine if the sum is direct for the following subspaces of ##\mathbf R^3## a) ## S = \{(x,y,z) \in \mathbf R^3 : x =z\}## ## T = \{(x,y,z) \in R^3 : z = 0\}## b) ## S = \{(x,y,z) \in \mathbf R^3 : x = y\}## ## T = \{(x,y,z) \in \mathbf R^3 ...

Hi all, I have the code below with a 'for' loop. I see the output is 17, the sum of all elements of the (variable?) result. I am a bit confused: I cannot tell which part of the code is used to determine that the values {1,2,10,4,5} must be added to each other? Please comment/correct: 1)We define...

Hello to everyone who's reading this. The problem I need help with is the following.: Homework Statement "Simplify to obtain minimum SOP. F(A, B, C, D) = A’B’CD’+AC’D’+ABC’+AB’C+AB’C+BC’D" The problem stated above has two provided solutions, the "main" one and the "alternate" one. I'm...

I have an issue/problem that relates to Bland initial treatment of external direct sums including Proposition 2.1.5 ... especially Bland's definition of the sum of a family of mappings ... Bland's text on this is as follows: In the above text by Bland we read the following: " ... ... We now...

I am new on this forum, this is my gift for you. Suppose ##(M_i)_{i \in I}## is a family of left ##R##-modules and ##M = \bigoplus_{i \in I} M_i## (external direct sum). Suppose ##N = \langle x_1, \cdots ,x_m \rangle## is a finitely generated submodule of ##M##. Then for each ##j = 1, \cdots...

Homework Statement S = 1+ x/1! +x2/2! +x3/3! +...+xn/n! To find S in simple terms. Homework Equations None The Attempt at a Solution I tried with Taylor's expansion, coshx and sinhx expansions. But cannot see consequence.

Hi, Lets say I have N independent, not necessarily identical, random variable. I define a new random variable as $$Y=Σ^{N}_{i=0} X_{i}$$ does Y follow a normalized probability distribution?

Suppose I have a matrix M = A + εB, where ε << 1. If A is invertible, under some assumptions I can write e Neumann series M-1 = (I - εA-1B)A-1 But if A is not invertible, how can I expand M-1 in powers of ε? Thanks in advance

Hello MHB! (Wave) A young man in high school I know has been essentially tasked with finding the following partial sum: $$S=\sum_{k=0}^{n}\left(\frac{2^k}{3^{2^k}+1}\right)$$ I honestly have no idea how to proceed, and I am hoping someone here can provide some insight. (Star)

How many times do we need to throw 2 dices to get more than 1/2 probability that at least once the sum of the two dices will be 12.

Homework Statement i have no idea how to proceed

Suppose $(M_i)_{i \in I}$ is a family of left $R$-modules and $M = \bigoplus_{i \in I} M_i$. Suppose $N = \langle x_1 \cdots x_m \rangle$ is a finitely generated submodule of $M$. Then for each $j = 1 \cdots m$, there is a finite $I_j \subset I$ such that $x_j \in \bigoplus_{i \in I_j} M_i$...

Homework Statement Calculate Σ (i=1, n) √i I want to write general formula, then use it for any n (like we have for Σ (i=1, n) i Homework Equations Σ (i=1, n) i = n (n+1) / 2 Σ (i=1, n) i^2 = n (n+1)(2n +1) / 6The Attempt at a Solution Comparing formulas provided above: I assume the answer...

Homework Statement The product of two positive numbers is 100. What numbers will produce the least possible sum? Confirm that the sum is in fact a minimum. Homework EquationsThe Attempt at a Solution For this question here I feel like the wording is a bit confusing, I tried my best please let...

Dear Everybody, I need some help with find M in the definition of the convergence for infinite series. The question ask, Prove that for $-1<r<1$, we have $\sum_{n=0}^{\infty} r^n=\frac{1}{1-r}$. Work: Let $\sum_{n=0}^{k} r^n=S_k$. Let $\varepsilon>0$, we must an $M\in\Bbb{N}$ such that $k\ge...

Homework Statement Calculate the sum of all hight-order bytes in array NUM1 and store the sum in a memory location named newH. Define newH as needed. Homework Equations - The Attempt at a Solution INCLUDE Irvine32.inc .data NUM1 sword 1h,2h,3h,4h,5h,6h,7h,8h,9h,10h,11h,12h,13h,14h,15h,16h...

##\sum_{n=0}^\infty \frac 1{p^{nz}}=1+\frac1{p^z}+\frac1{p^{2z}}+\frac1{p^{3z}}...## ##\frac 1{p^z}\sum_{n=0}^\infty \frac 1{p^{nz}}=\frac1{p^z}+\frac1{p^{2z}}+\frac1{p^{3z}}+\frac1{p^{4z}}...## something happens and it shows: ##\frac 1{p^z}\sum_{n=0}^\infty \frac...

I´m not sure, whether this little challenge has been posted before. I have searched the forum and didn´t find it. It might still be a duplicate though ... Find the sum of fractions $$\frac{2}{3\cdot5}+\frac{2\cdot4}{3\cdot5\cdot7}+\frac{2\cdot4\cdot6}{3\cdot5\cdot7\cdot9}+...$$

The problem I'd like to calculate the value of this sum: $$3 \sum^\infty_{k=1}\frac{1}{2k^2-k}$$The attempt ## 3 \sum^\infty_{k=1}\frac{1}{2k^2-k} = [k=t/2] = 3 \sum^\infty_{t=2}\frac{1}{2 \left( \frac{t}{2} \right)^2-\frac{t}{2}} = 3 \sum^\infty_{t=2}\frac{1}{ \frac{t^2}{2} - \frac{t}{2}} = 3...

If I have an asset that has a 10% chance to fail and I have ten of these assets in a basket, then what is the chance that one will fail in this basket? 10%?:partytime: What is the chance of 10 failing? 0,01%? Please also explain in some laymans terms. I am a total noob when it comes to...

So the first term of the Lagrangian is proportional to ##{F_{\mu \nu}}{F^{\mu \nu}}##. I wrote out the matrices for ##{F_{\mu \nu}}## and ##{F^{\mu \nu}}## and multiplied at the terms together and added them up, but some of the terms didn't cancel like they should have. Should I have used minus...

Hi! (Not sure which forum to pick for this question. This looked like the best one. I apologies if it is not) I have a number of functions (say m functions) with integer domain. All functions are increasing. (Increasing in the sense not decreasing, $f(n+1) \ge f(n)$.) I want an algorithm to...

Suppose I have a 1-D harmonic oscilator with angular velocity ##\omega## and eigenstates ##|j>## and let the state at ##t=0## be given by ##|\Psi(0)>##. We write ##\Psi(t) = \hat{U}(t)\Psi(0)##. Write ##\hat{U}(t)## as sum over energy eigenstates. I've previously shown that ##\hat{H} = \sum_j...

Find the exact sum of the series: $$S = \frac{1}{1\cdot 2\cdot 3\cdot 4}+\frac{1}{5 \cdot 6 \cdot 7 \cdot 8}+...$$