Sum Definition and 1000 Threads

Some popular math videos point out that, for example, the value of -1/12 for the divergent sum 1 + 2 + 3 + 4 ... can be found by integrating n/2(n+1) from -1 to 0. We can easily verify a similar result for the sum of k^2, k^3 and so on. Is there an elementary way to connect this with the more...

In this code, I define a function of x as the sum of the first x integers. In[7]:= fnSum[x_] := Sum[k, {k, 1, x}] In[8]:= fnSum[x] Out[8]= 1/2 x (1 + x) In[9]:= fnSum[3.5] Out[9]= 6 I would like now to take the symbolic formula underlying fnSum, and use it with real arguments. How can I...

Summary:: i get a proof that sum of rational and irrational is rational which is wrong(obviously) let a be irrational and q is rational. prove that a+q is irrational. i already searched in the web for the correct proof but i can't seem to understand why my proof is false. my proof: as you...

please help me

Apologies if this isn't the right forum for this. In my stats homework we have to prove that the expected value of aX and bY is aE[X]+bE[Y] where X and Y are random variables and a and b are constants. I have come across this proof but I'm a little rusty with summations. How is the jump from the...

I'm trying verify the proof of the sum rule for the one-dimensional harmonic oscillator: $$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$ The exercise explicitly says to use laddle operators and to express $p$ with $$b=\sqrt{\frac {mw}{2 \hbar}}-\frac...

In spite of all the problems that apparently arise from my questions or from what these questions represent (among these problems 'do not seem to agree with the underlying framework'), i would be obliged if someone can answer me the following questions, which i am hesitant to ask here as a...

Show that $3^{2008}+4^{2009}$ can be written as a product of two positive integers each of which is larger than $2009^{182}$.

Question 1; Method 1 If the sum of the first four terms is 139 then S4=139 139=1/2(4)(2a+(4-1)d) 139=2(2a+3d) 139=4a+6d----- [1] The part of this question that is confusing is the "the sum of the next four terms is 115". Would this mean that S8=S4+115=139+115=254? In which case...

Usually, I saw that string theory (perturbative, or matrix models) are made in a fixed background. Even if you consider that the metric is quantized and etc. there is an apparent physically motivated need for making a sum over topologies (manifolds, conifolds, orbifolds, and etc), for example...

Why the summation of the following function will be canceled out when we take the partial derivative with respect to the x_i? Notice that x_i is the sub of (i), which is the same lower limit of the summation! Can someone, please explain in details?

Evaluate $\dfrac{1}{1-\cos \dfrac{\pi}{9}}+\dfrac{1}{1-\cos \dfrac{5\pi}{9}}+\dfrac{1}{1-\cos \dfrac{7\pi}{9}}$.

Seems to me the answer is a specific vector: The second forms a plane, while the first X is just a vector. The intersection between the λX that generates the (properties of all vectors that lie in the...) plane (i am not saying X is the director vector!) How to write this in vector language?

##\sum_n \frac{1}{n^c}## converges for ##c\gt 1##. Is there an expression for the value of the sum as a function of ##c##?

for the sum, ##\frac {1}{∝^3}##+##\frac {1}{β^3}##=##\frac {β^3+∝^3}{∝^3β^3}## =##\frac {(∝+β)[(∝+β)^2-3∝β]}{∝^3β^3}## =##\frac {-b}{a}##...

I have found code to find simply the minimum numbers needed, but I need to do it without repetition given the nature of an electric circuit. I hope that is a sufficient enough explanation of the problem. Despite being an engineering project this aspect is more mathematical.

We're given a function which is defined as : $$ f:[0,1] \mapsto \mathbb R\\ f(x)= \begin{cases} x& \text{if x is rational} \\ 0 & \text{if x is irrational} \\ \end{cases} $$ Let ##M_i = sup \{f(x) : x \in [x_{i-1}, x_i]\}##. Then for a partition ##P= \{x_0, x_1 ...

Physicists Stephen W Hawking and James B Hartle 1 proposed that the universe, in its origins, had no boundary conditions both in space and time. To do that, they proposed a sum over all compact euclidean compact metrics. I have heard that they only considered these metrics in order to simplify...

There is a paper here: https://www.mdpi.com/1099-4300/19/5/188 And a lengthy article here: https://www.quantamagazine.org/a-theory-of-reality-as-more-than-the-sum-of-its-parts-20170601/ The general argument concerns causal emergence and whether all causal agency arises directly from the micro...

since the first term is ##g(0)= \frac {1}{3}## & last term is ##g(1)=\frac {4}{6}## it follows that the ##\sum_{0}^1 g(x)##= ##\frac {1}{3}##+##\frac {4}{6}=1## is this correct?

#include<stdio.h> #include<math.h> double foo(int n){ if(n==1){ return(1); } if(n!=0){ return( sqrt((n)+foo(n-1) ) ); } } int main(){ int num; printf("Enter the number: "); scanf("%d",&num); foo(num); printf(" %lf ",foo(num)); return(0); }I...

I tried to write it as n^2/2^n (1+1/2n)^n But I am stuck there and don't know what to try next.Thanks for any help in advance!

Balance the forces east-west: 3.5kg*sin45º + 4.2kg*cos30º - 4.8kg*sin30º - E*cosΘ = 0 E*cosΘ = 3.712 kg balance north-south: 2.8kg + 3.5kg*cos45º - 4.2kg*sin30º - 4.8kg*cos30º + E*sinΘ = 0 E*sinΘ = 0.982 EsinΘ / EcosΘ = 0.982 / 3.712 tanΘ = 0.2645 Θ = 14.8º ◄ E = 3.712kg / cos14.8º = 3.84...

3.7.4. The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares? $x+y=16\implies y=16-x$ Then $x^2+(16-x)^2=2 x^2 - 32x + 256$ So far ... Hopefully

1. Is it because the initial formula start the series from ##n = 2##? 2. If the initial formula is used, can I find ##S##, which $$S=\lim_{n\to\infty} \frac{2}{n^2-1}=\frac{2}{\infty}=0$$? Why that answer is different if the formula is changed.

S8.3.7.3. Find two positive numbers whose product is 100 and whose sum is a minimum $x(100-x)=100x-x^2=100$ So far Looks like it's 10+10=20Doing all my lockdown homework here since I have no access to WiFi and a PC. and just a tablet where overkeaf does not work

r_{13}=r_{23}=\sqrt{(30*10^{-3})^2+(90*10^{-3})^2}=\sqrt{9*10^{-3}}\\ F^E_{13}=F^E_{23}=9E9\cdot\frac{5*10^{-9}\cdot3*10^{-9}}{9*10^{-3}}=1.5*10^{-5}\\ \theta=tan^{-1}(\frac{90*10^{-3}}{30*10^{-3}})=71.565\,degrees\\ \vec{F}^E_{13}=<F^E_{13}cos\theta, F^E_{13}sin\theta> = <4.743*10^{-6}...

I have calculate a serie of view factors for a given geometry and its sum is aproximately one but not exactly. My values are: 0,1134 0,1307 0,2446 0,12393 0,115053 0,010084 0,007334 0,1071 0,0145 0,0128 0,0919 0,01675 0,00463 0,00344 The sum now is equal...

Hello all, In high school physics, the magnitude sum of vector addition can be found by cosine rule: $$\vec {R^2} = \vec {F^2_1} + \vec {F^2_2} + 2 \cdot \vec F_1 \cdot \vec F_2 \cdot cos ~ \alpha$$ and its angle are calculated by sine rule: $$\frac {\vec R} {sin ~ \alpha} = \frac {\vec F_1}...

ok I don't don't know de jure on this so ... is it just plug and play?? find factors of -48 $-1(48)=-48$ $-2(24)=-48$ $-3(16)=-48$ $-4(12)=-48$ $-6(8)=-48$ check sums for positive number $-1+48=47$ $-2+24=22$ $-3+16=13$ $-4+12=8$ $-6+8=2$it looks like c. 5

I am not sure what I can do with the equation. I realize that ## \vert c_1 \vert ^2 = \vert c_2 \vert ^2 = \frac{1}{2} ## does not mean that ## c_1 ^2 = c_2 ^2 = \frac{1}{2} ## or that ## c_1 = c_2 ##, so I don't know how to use it. I think ideally I might have something like ##P = \vert c_1...

Suppose the Bell operator ##B=|AB(1,2)+AB(1,3)+AB(2,3)|## With ##AB\in{1,-1}## Nonlocal realism implies ##B\in{1,3}## However using usual matrix sum one eigenvalues for the result of measurement can be smaller than 1, implying nonlocal realism cannot explain the quantum result. However if...

e.g Can we write it as $$f(a)+f(a+dx)+f(2a+dx)+f(3a+dx)+...f(b)=\int^b_a f(x)dx$$...(?) Although $$\int f(x)dx$$ given the area tracked by thr function with the x-axis between a and b Thanks.

The sum of ten integers is 0. Show that the sum of the fifth powers of these numbers is divisible by 5. For this one I don't know what I have to do at all other than brute-forcing which may even be impossible.

In A.P. French's Special relativity the author said, The mass and length of the box are irrelevant here. He said the momentum of the radiation is ##E_{radiation}/c##. We know that the momentum of a single photon with energy ##E_{photon}## is ##p_{photon}=E_{photon}/c##. So is...

Hi all; I have a very basic understanding of sequences and series and recently encountered a sequence which really has me confused: $$(\frac{1}{5}+(\frac{1}{5}+(\frac{1}{5}+(...)^2 )^2)^2)^2$$ What type of sequence would you call this? I couldn't even google it because I couldn't work out how to...

Consider the following series with the following pattern $$\frac {1}{1×3}+\frac {1}{5×7}+\frac {1}{9×11}...$$ How would you go about working out what the general rule for this sum is? That is in the form of ##\sum_{n=a}^{b}f(n)## Any help is greatly appreciated.

I got answer to (a), which is 3/4 sin thteta - sin ((3^(n+1)) theta) / (4 . 3^n) but I do not know how to use this result to prove next question. I tried to change theta into pi/2 - theta so that sin change to cos or vice versa but not working. Thanks

Hey! :o Let $n\in \mathbb{N}$, $2\leq m\in \mathbb{N}$ and $a\in \mathbb{Z}$. I want to show that $a\left (m+1\right )^n \overset{(9)}{\equiv} a$. I have done the following: \begin{equation*}a\left (m+1\right )^n \overset{(9)}{\equiv} a\left (0+1\right )^n \overset{(9)}{\equiv} a\cdot 1^n...

If I have a sum ##f(x) + g(x) = c##, with ##c## a constant, does this imply that both ##f(x)## and ##g(x)## are also constants? If I just solve this equation for ##x##, I will find some values of ##x## which satisfy the equation. However, if I require that the equation be true for all ##x##...

In physics we often change a sum to an integral.But I am not clear when can we change a sum to an integral?When a term of sum is comparable to the sum,can we change the sum to integral?

ok basically t is 3 hours appart except between 7 and 12 of which I didn't know if we should intemperate. other wise it is just adding up the 4 $(t)\cdot(R(t))$s.

Hi, this is my try:Thanks.

Example done in class: The problem and my solution: My solution seems incorrect because if I try to plug in 0, I don't get the initial condition given in the problem. Does anyone see what I've done wrong along the way? Thanks.

First, I got rid of amperemeters with 0 values. These are 9. 11 and 12. Amperemeter 4 will show the maximum value of electric current as it is placed directly between E and F. But how to know its value? Will it be 18 mA? I doubt because 18 mA is not said to be the maximum value. All other...

Please see below my attempt to perform the convolution operation on two discrete-time signals as part of my Digital Signal Processing class. I suspect my folding operation, i.e. flipping one signal about k=0, might be the cause. Ostensibly the answer of the convolution sum evaluated at n=-2...

I've found that ##N_1## is 1. But it's really tiresome to find them one by one. I also tried to use the equation but couldn't. Please help me out.

Solution to the problem tells us that ##S_5 + i S_6## is the sum of the terms of a geometric sequence and thus the solutions should be : $$S_5 = \frac{\sin( (n+1) x)}{\cos^n(x) \sin(x)},\,\,\,\, S_6 = \frac{\cos^{n+1}(x) - \cos((n+1)x)}{\cos^n(x) \sin(x)} , x \notin \frac{\pi}{2} \mathbb{Z}$$...

Hi, I have a particular equation in a paper, wherein the author specifies an infinite series. The author has apparently found the sum of the series and calculated the equation. Can anyone please help me in understanding how to sum such a series. I have attached part of the paper with the...

I'm using the sum of a geometric series formula, but I'm not sure how to find the ratio, r. The n is confusing me. The solution is below, but I'm having trouble with the penultimate step.