Sum Definition and 1000 Threads

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...

can you help me solve this just using one variable the sum of two numbers is 9 and their difference is 6. what are the numbers? thanks!

I can't wrap my head around this proof that the sum of two nilpotent ideals is nilpotent, I get stuck at one stage: http://imageshack.com/a/img706/5732/5wgq.png I'm fine with every except showing by induction (I+J)^{N+k} = I^k \cap J + I \cap J^k . Here's my attempt; Base case: k =...

How to find the sum using complex analysis $$sin^3x+sin^32x+sin^33x+sin^34x+...+sin^3nx$$

If the sum of the surfaces of a cube and a sphere as is constant, deierminar the minion of the diameter of the sphere to the edge of the cube in cases in which: 272) The sum of the volumes is minimal 273) The sum of the volumes is maximum And the answer are 272 = 1 and 273 = infinit Ok Vs =...

I have a finite sum of the form: ∑n=1Nexp(an+b√(n)) Is there any trick to evalute this sum to a closed form expression? e.g. like when a finite geometric series is evaluated in closed form.

If I'm given a function ##f(x) = A cos (x) + B sin (x)##, is there any way to turn this into an expression of the form ##F(x) = C e^{i(x + \phi)}##? I know how to use Euler's formula to turn this into ## \alpha e^{i(x + \phi)} + \beta e^{-i(x + \phi)}##, but is there a way to incorporate the...

Given 2 subspaces and a linear operator, is the image of the direct sum of the subspaces equal to the union of the images under the operator? Thanks

The attached pdf shows integral approximations of two sums, which are done in my book. In the first there is no result but the book simply states that one can approximate the sum by an integral. My question is: How is this done? Normally when you approximate a sum by an integral you have a sum...

Homework Statement As the title indicates. I'm given two independent exponential distributions with means of 10 and 20. I need to calculate the probability that the sum of a point from each of the distributions is greater than 30. Homework Equations X is Exp(10) Y is Exp(20) f(x) =...

Homework Statement Given two 2-dimensional vectors \overline{a} and \overline{b} of moduli l\overline{a}l = 3u and l\overline{b}l = 4u, and forming an angle  of 120 degrees between them, determine the modulus of the sum vector \overline{s} = \overline{a} + \overline{b} and the angle between...

Homework Statement Consider the sequence ak = (3)/(4^(2k)). Show is convergent, find sum. Please check work. Homework Equations ak = 3/(4^2k) let s {n} be the series associated with the sequence. Cannot write summation notation here, but k starts at 1 (k = 1 on bottom) and infinity...

Homework Statement Prove the following result: \frac{1}{2.2} + \frac{ 1}{3.2^2} + \frac{1}{4.2^3} ... = 2ln2 -1 Homework Equations The Attempt at a SolutionI tried writing down the nth term of the series which is 1/(n+1)2^n But don't know where to move after this.

Homework Statement X is uniform [e,f] and Y is uniform [g,h] find the pdf of Z=X+Y Homework Equations f_z (t) = f_x (x) f_y (t-x) ie convolution The Attempt at a Solution Obviously the lower pound is e+g and the upper bound is f+h so it is a triangle from e+g to f+h...

Homework Statement Sum to infinity \frac{1}{2!} - \frac{ \pi ^2}{4^2.4!} + \frac{\pi^4}{4^4.6!} ... Homework Equations The Attempt at a Solution I thought the series was similar to the Maclaurin expansion of cos x so I tried putting in x= ∏/4 But I end up with the...

Homework Statement Prove the following result: \frac{1}{1!} + \frac{1+2}{2!} + \frac {1+2+2^2}{3!} ... = e2 - e Homework Equations The Attempt at a Solution Could someone please give me a hint on what to do . I tried writing out the maclaurin series for e^2 and e but...

A professor in Cambridge University has showed his proof that the sum of natural number was equal to -1/12. The video can be found on the internet.Well, although the way to proof I think is really ridiculous, it could be a good way to building a new math model. Since we think the sum of natural...