Sum Definition and 1000 Threads

By considering the product of complex numbers: z = (2+i)(3+i) Show that \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \frac{\pi}{4}

dear all, let's consider a length L and divide it into N number of small segments uniformly. Then the length of every segment should be L/N. then we add these segments up, which is L=\sum\frac{L}{N} then we take the limit N→\infty at both sides, this means...

Hi! I'm taking my first course in statistics and am hoping to get some intuition for this set of problems... Suppose I have a bowl of marbles that each weighs m_{marble}=0.01 kg. For each marble I swallow, there is a chance p=0.53 that it adds m_{marble} to my weight, and chance 1-p that...

I was looking at a homework question posted here requiring proof that the vectors from the vertices of a triangle to the midpoint of the opposite edge sum to zero, and it struck me that there is a more general property: Consider a set of points, \{A_0, A_1, \ldots A_n\}. The midpoint of...

Hi! Many students know that A\sin(x) + B\cos(x) =\sqrt{A^2+B^2} \sin{(x+\arctan \frac{B}{A})}. I have seen just one deduction of that relation, showed by set up a system of two equations, solving for amplitude and phase shift. Is it possible to deduce the relation in a vectorial way, or in...

Homework Statement I have been sitting here for the last hour trying to figure it out but I can't seem to be able to find what I'm doing wrong. I need to expand an equation. Homework Equations a2 - a - 3 The Attempt at a Solution a2 - 1a - 3 The product is -3 and the sum -1...

Homework Statement I have to find a natural number N that satisfies this equation: \sum^{N}_{i=1} \frac{1}{i} > 100 Homework Equations I tried finding a close form of the sum but couldn't find anything useful. The Attempt at a Solution Well after trying some numbers in maple I...

Given $a,\,b,\,c,\,d$ are real numbers such that $a+b+c+d=5$ $2a+4b+8c+16d=7$ $3a+9b+27c+81d=11$ $4a+16b+64c+256d=1$ Evaluate $5a+25b+125c+625d$.

Find the sum of all positive integers $a$ such that $\sqrt{\sqrt{(a+500)^2-250000}-a}$ is an integer.

Homework Statement Find an N so that ##∑^{\infty}_{n=1}\frac{log(n)}{n^2}## is between ##∑^{N}_{n=1}\frac{log (n)}{n^2}## and ##∑^{N}_{n=1}\frac{log(n)}{n^2}+0.005.## Homework Equations Definite integration The Attempt at a Solution I began by taking a definite integral...

Let $k_n$ denote the integer closest to $\sqrt{n}$. Evaluate the sum $\dfrac{1}{k_1}+\dfrac{1}{k_2}+\cdots+\dfrac{1}{k_{1980}}$.

Homework Statement Compute \sum\frac{4}{(-3)^n}-\frac{3}{3^n} as n begins from 0 and approaches infinity Homework Equations The Attempt at a Solution I'm just getting started on sequences and series, and so far learned about the limit test, comparison test, arithmetic / geometric...

Homework Statement Prove that if bonding-pair repulsions were maximized in CH3X, then the sum of the bond angles would be 450°. Homework Equations In a perfect tetrahedral molecule (e.g. methane), the sum of the bond angles is about 438 degrees (109.5° times 4). The Attempt at a...

Homework Statement Find the value of An and given that f(x) = 1 for 0 < x < L/2, find the sum of the infinite series. Homework Equations The Attempt at a Solution The basis is chosen to be ##c_n = \sqrt{\frac{2}{L}}cos (\frac{n\pi }{L}x)## for cosine, and ##s_n = \sqrt{\frac{2}{L}}sin...

Problem: If $0<x<1$ and $$A_n=\frac{x}{1-x^2}+\frac{x^2}{1-x^4}+\cdots +\frac{x^{2^n}}{1-x^{2^{n+1}}}$$ then find $\displaystyle \lim_{n\rightarrow \infty}A_n$. Attempt: I tried to see if it can be converted to a telescoping series but I had no luck. Then, I tried this: $$\lim_{n\rightarrow...

Problem: Evaluate $$\mathop{\sum \sum}_{0\leq i<j\leq n} (-1)^{i-j+1}{n\choose i}{n\choose j}$$ Attempt: I wrote the sum as: $$\sum_{j=1}^{n} \sum_{i=0}^{j-1} (-1)^{i-j+1}{n\choose i}{n\choose j}$$ I am not sure how to proceed from here. I tried writing down a few terms but that doesn't seem...

For the question, shouldn't the sum be a(1/1-r) since we know lrl < 1 then that rn → 0 as n → ∞? I just don't quite understand why they wrote the sum is a(r/1-r). Is there a specific reason they did this? This is just a regular geometric series right? Is there any difference since the sum starts...

Homework Statement Find the sum of the first n terms of the sequence U1, U2, U3... Ur Homework Equations The Attempt at a Solution $$ \sum_{r = 1}^n (1 + (-1)^r) = n + (-1)^n $$ But I don't this is right... any help?

Homework Statement Wn = 2 + 3(1/2)^n Homework Equations The Attempt at a Solution I am confused, all I tried so far is writing out the first 5 terms, but all that was helping me to do is basically find the Sum to infinity... so what should I do to find the Sum to n terms? I know...

Hi! Consider the function \frac{d^n}{dx^n} \sum_{k=1}^m \sin{kx}, \quad 0 \leq x \leq \pi/2 . If n is odd this function attains its largest value, \sum_{k=1}^m k^n at x=0 . But what about if n is even? Where does the maximum occur and what value does it take? Any help is much...

I am reading James Munkres' book, Elements of Algebraic Topology. Theorem 6.5 on page 39 concerns the homology groups of the connected sum of two projected planes. Munkres demonstrates the following: H_1 ( P^2 \# P^2 ) \simeq \mathbb{Z} \oplus \mathbb{Z} / 2 ... ... ... (1) and H_2 ( P^2...

I just completed a post in the Topology and Advanced Geometry forum regarding the connected sum of two projective planes. I wanted to use the symbol # for the connected sum as is usual in the topology books I am studying - but just typing in the symbol 'upsets' latex and so my post cannot be...

Here is the question: I have posted a link there to this question so the OP can view my work.

Homework Statement find the sum of the following series: \sum_{n=1}^\infty nx^{n-1} , |x|<1 Homework Equations \frac{a}{1-r} The Attempt at a Solution i know that a function representation for that series is -\frac{1}{(1-x)^2} but how is it possible to find the sum of a series with a...

Homework Statement I understand this BCD sum circuit. The only thing that I'm not understanding is why last full adder carry out also triggers an 6-sum in the other part of the circuit. I mean, if the nibble is not an valid BCD number, we sum six to the number. Not valid BCD numbers are...

My mind has gone blank and I've suddenly forgotten basic algebra, please could someone give me direction on how to make P the subject of this equation? E = (P^2 C^2 + M^2 C^4)^1/2 + (P^2 C^2)^1/2 thanks for any help

Find the sum of the positive solutions to $5+x\lfloor x \rfloor-2x^2=0$.

Evaluate $\displaystyle\lower0.5ex{\mathop{\large \sum}_{n=2009}^{\infty}} \dfrac{1}{n \choose 2009}$.

Hi guys, I have this general question. If we are asked to show that the direct sum of ##U+W=V##where ##U## and ##W## are subspaces of ##V=\mathbb{R}^{n}##, would it be possible for us to do so by showing that the generators of the ##U## and ##W## span ##V##? Afterwards we show that their...

If we consider a bipartite system as in EPRB experiment we get the probabilities : p(++)=p(--)=1/4*(1-cos(theta)) p(+-)=p(-+)=1/4*(1+cos(theta)) p(+A)=p(+B)=p(-A)=p(-B)=1/2 Thus the sum of all the probabilities equals 3... How does that come ? Is it because in fact there are only...

I'm not sure which category this post actually belongs to, or if the title of this post is even accurate. I guessed Calculus was the closest one. I watched this video on the web after a professor told me this mathematical phenomenon (http://www.youtube.com/watch?v=w-I6XTVZXww‎). It asserts...

How is it exactly i convert between a k-space sum an integral? Assume that we have some macroscopic solid. Periodic boundary conditions leads to kx,ky,kz = 2π/L, so each k-space state fills a volume (2π/L)3 or has a density of V/(2π)3. To then count for instance the number of state with...

Given two coefficient binomials \binom{a}{b} and \binom{c}{d} is possbile to express the sum and product those coefficient binomials as one other?

find a point on the axis OX whose sum of distances to landmarks: (2, 0) and (0, 3) is minimal. Answer (2,0) As the title says it is the same as the former So the equations must be sqrt((x-2)2+02)+sqrt((x-0)2+ 9) because the point is (x.0) but it seems I am wrong because i don't get the answer

I am trying to derive the distribution for the sum of two random vectors, such that: \begin{align} X &= L_1 \cos \Theta_1 + L_2 \cos \Theta_2 \\ Y &= L_1 \sin \Theta_1 + L_2 \sin \Theta_2 \end{align} With: \begin{align} L_1 &\sim \mathcal{U}(0,m_1) \\ L_2 &\sim...

Homework Statement Which is equivalent to: cos(∏/2 + x) - cos(∏/2 - x)? A) -2cos(x) B) -2 C) 0 D)-2sin(x) Homework Equations Cos (A-B) The Attempt at a Solution I am totally stuck :( please help!

Prove the following \sum_{n=1}^\infty \sum_{m=1}^\infty\frac{a^{n+m}}{(n+m)!} = ae^a-e^a+1

Homework Statement If f(r) = r^3-5r^2+6r then \sum_{r=4}^{\infty } \dfrac{1}{f(r)} is The Attempt at a Solution I could decompose the above summation into something like this \dfrac{1}{3} \left( \sum \dfrac{1}{(r-2)(r-3)} - \sum \dfrac{1}{r(r-2)} \right) But from here I'm not...

find a point on the axis OX whose sum of distances to landmarks: (1, 2) and (4, 3) is minimal. Answer (2,0) (x-1)2+(y-2)^2 + (x-4)2+(y-3)^2 = D Y = mx I don't know if solve each distance apart or how i wrote??

Homework Statement Determine whether the series is either Converge or Diverge, if it's convergent, find its sum ∑ from n=1 to ∞ of (1+2^n)/(3^n) Homework Equations The Attempt at a Solution Steps: 1) i replaced the (1+2^n) to just (2^n) and my equation behaves like ∑ (2/3)^n which...

Hi PF. This a fact well aware to just about anyone that has had even basic chemistry, but I'm having a hard time coming to an understanding as to why this must be true. So why? Also, if I knew that some box contained, say, a proton and an electron, could I ever know whether or not, inside...

Homework Statement Hi Everyone, I am trying to show why the given sum is zero. I am pretty sure it is zero. Homework Equations sin[8*\pi*n/5]+sin[12*\pi*n/5] n is an integer. The Attempt at a Solution n----sin[8*\pi*n/5]----sin[12*\pi*n/5] 0 ----...

Suppose we have a Markov chain with stationary distributions ##p_n=\frac{a}{nb+c}p_{n-1}## for ##n\in\mathbb{N}## where ##a,b## and ##c## are some positive constants. It follows that ##p_n=p_0\prod_{i=1}^n\frac{a}{ib+c}##. Normalisation yields...

Homework Statement Find the sum of the series: ∞n=1∑(6)/((2n-1)(2n+1)) Homework Equations The Attempt at a Solution S1=2 S2=2+(6/15) S3=2+(6/15)+(6/35) This is the part where I get a little confused. It looks like the denominator is getting bigger... So does it approach...

Homework Statement Consider, in the circuit from the image, i1(t) = 5 cos(2t + 10º) v1(t) = 10 cos(2t - 60º). Find the value of the current ix(t). Options given: A: ix(t) = 9.9 cos(2t - 129.2º) B: ix(t) = 9 cos(2t - 29.2º) C: ix(t) = 99 cos(2t + 129.2º) D: ix(t) = 0.99 cos(2t +...

Evaluate $\dfrac{1}{3^2+1}+\dfrac{1}{4^2+2}+\dfrac{1}{5^2+3}+\cdots$

Show that for an odd integer $m\ge 5$, $\displaystyle {m\choose 0} 5^{m-1}-{m\choose 1} 5^{m-2}+{m\choose 2} 5^{m-3}-\cdots+{m\choose m-1} $ is not a prime number.

Homework Statement a) Show that the Fourier Cosine Series of f(x)=x,\quad 0\leq x<L is x ~ \frac{L}{2}-\frac{4 L}{\pi ^2}\left[\left(\frac{\pi x}{L}\right)+ \frac{\cos\left(\frac{3\pi x}{L}\right)}{3^2}+\frac{\cos\left(\frac{5 \pi x}{L}\right)}{5^2}+\dots\right] b) use the above series to...

Homework Statement From Mary Boas: Math for Phys. Sci. Ch1.15.20 20. By computer or tables, find the exact sum of each of the following series. a. \sum _{ n=1 }^{ \infty }{ \frac { { n } }{ { (4{ n }^{ 2 }-1) }^{ 2 } } } Homework Equations N/A. One is supposed to use an analytic...

Hello! (Wave) I am looking at the proof that if $f$ is integrable and $k \in \mathbb{R}$,then $kf$ is also integrable and $\int_a^b{(kf)}=k \int_a^b{f}$. The following identity is used at my textbook: $$U(kf,P)=\left\{\begin{matrix} k \cdot U(f,P), \text{ if } k>0\\ k \cdot L(f,P), \text{ if...