Series Definition and 998 Threads

From my physical problem, I ended up having a sum that looks like the following. S_N(\omega) = \sum_{q = 1}^{N-1} \left(1 - \frac{q}{N}\right) \exp{\left(-\frac{q^2\sigma^2}{2}\right)} \cos{\left(\left(\mu - \omega\right)q\right)} I want to know what is the sum when N \to \infty. Here...

For what values of β the following series converges $$ \sum_{k=1}^{\infty} k^{\beta} \left( \frac{1}{\sqrt k} - \frac{1}{\sqrt {k+1}}\right) $$ I thought of doing it like this $$ \frac{k^{\beta} }{\sqrt k} - \frac{k^{\beta} }{\sqrt{k+1}}$$ $$0 \lt \frac{k^{\beta} }{\sqrt k} - \frac{k^{\beta}...

##\sum _{n=0}^{\infty }\:\frac{sin\left(2^nx\right)}{2^n}## I have to show the series is periodic, ##2\pi ## ( I think ), it is related to fourier - analyze fourrier course at academics, so I might be right, or just periodic, we learned only periodic of ## 2\pi ## ( of course also continuous...

Hi, a question regarding something I could not really understand The question is: Let V be a space with Norm $||*||$ Prove if $v_n$ converges to vector $v$. and if $v_n$ converges to vector $w$ so $v=w$ and show it by defintion. The question is simple, the thing I dont understand, what...

Hi everyone! It's about the following task: show the convergence or divergence of the following series (combine estimates and criteria). I am not sure if I have solved the problem correctly. Can you guys help me? Is there anything I need to correct? I look forward to your feedback.

I've occasionally seen examples where autonomous ODE are solved via a power series. I'm wondering: can you also find a Taylor series solution for a non-autonomous case, like ##y'(t) = f(t)y(t)##?

My textbook solution states that 1 & 2 are in parallel and so is 3 & 4 and those 2 are in series. That is, (1 P 2) S (3 P 4). My thinking is such: points A & B are of same potential, say V, C & D are of same potential, say x and E & F are are of same potential, say 0. So I can say that 1 and 3...

In the "Inspector Montalbano" TV Series, the characters often drink coffee in small cups and it is poured from a small silver colored container. I've read on the internet that typical Sicilian coffee is expresso. Is the series consistent with that? Do homes in Sicily usually have expresso...

We're given the series ##\sum_{n=1}^{\infty} [ \sqrt{n+1} - \sqrt{n} ]##. ##s_n = \sqrt{n+1} - 1## ##s_n## is, of course, an increasing sequence, and unbounded, given any ##M \gt 0##, we have ##N = M^2 +2M## such that ##n \gt N \implies s_n \gt M##. Thus, the series must be divergent. But...

(I need help with the 2nd part as I can answer the theory part properly). For E=4 eV we can find the wavelength of emitted photon. E= 4 eV = 6.4087e-19 J Using E= hc/λ we get λ=310 nm (approx) My doubt is that this should fall in the Balmer Series but we know that the lowest wavelength value...

##s_1=2## ##s_2=4## ##s_3=5.333## ##s_4=5.9999## ##(s_n)## is increasing, but unable to guess a bound. Let's see if Cauchy criterion can do something. For n>2, $$ s_{n+k} - s_n = \frac{2^{n+1} }{(n+1)!} + \frac{ 2^{n+2} }{(n+2)!} + \cdots \frac{2^{n+k} }{(n+k)!} $$ $$ s_{n+k} - s_n <...

I am currently looking at section IIA of the following paper: https://arxiv.org/pdf/gr-qc/0511111.pdf. Eq. (2.5) proposes an ansatz to solve the spheroidal wave equation (2.1). This equation is $$ \dfrac{d}{dx} \left((1-x^2) \dfrac{d}{dx}S_{lm} \right) + \left(c^2x^2 + A_{lm} -...

First I got ##f(0)=0##, Then I got ##f'(x)(0)=\frac{\cos x(2+\cosh x)-\sin x\sinh x}{(2+\cosh x)^2}=1/3## But when I tried to got ##f''(x)## and ##f'''(x)##, I felt that's terrible, If there's some easy way to get the anwser?

My first attempt was ##... + n^{2} + (n+1)^{2} > \frac {1}{3} n^{3} + (n+1)^{2}## then we must show that ##\frac {1}{3} n^{3} + (n+1)^{2} > \frac {1}{3} (n+1)^{3}## We evaluate both sides and see that the LHS is indeed bigger than RHS. However, this solution is inconsistent so I am asking for...

Greetings According to my understanding: if x converges in 4 means that the series converges -1<x+3<7 but the solution says C Any hint? thank you!

I do not have any reasonable attempts at this problem, as I am trying to figure out how one can get the correct answer when we are not given any values. Maybe if some of you sees a mistake here, that implies that the values from the previous example should be used... ##a_3 = a_1 \cdot k{2}##...

I have tried inserting 0.955 in the above formula for the sum of a geometric series and setting it equal to 3,000,000 (S_n) with n =3. This did not work out well My second attempt was, considering that the payment is paid every year in the future, to use the convergence formula. There k = 0.955...

I am trying to understand the series expansion of $$(1-cx)^{1/x}$$ The wolframalpha seems to solve the problem by using taylor series for ## x\rightarrow 0## and Puiseux series for ##x\rightarrow \infty##. Any ideas how can I calculate them ...

Hi, I'm writting because I sort of had an idea that looks that it should work but, I did not find any paper talking about it. I was thinking about approximating something like algebraic functions. That is to say, a function of a complex variable z,(probably multivalued) that obeys something...

A convergent version ( i.e. convergent in the critical strip) of the traditional series for the Riemann Zeta is derived in the video linked at the bottom. It gives the correct numerical values (at least along the critical line, where I tried it out). But although it works numerically, I'm...

I believe I am doing everything right up until the point where I have to try and find a recurrence relation. I honestly have no idea what to do from there. I've listed my work in getting the powers of n and the indicies to all match. Any help appreciated. Here is the original DE...

Dear Colleagues I hope this post belongs here in calculus. It concerns a finite series for which I am seeking the sum. I tried using MATHEMATICA which didn't accept it. Perhaps if someone has Maple or any other software who can do it. Here it is attached. I shall be most grateful

Hello, I am reading a course on signal processing involving the Z-transform, and I just read something that leaves me confused. Let ##F(z)## be the given Z-transform of a numerical function ##f[n]## (discrete amplitudes, discrete variable), which has a positive semi-finite support and finite...

Find question and solution here Part (i) is clear to me as they made use of, $$\sum_{r=n+1}^{2n} u_r=\sum_{r=1}^{2n} u_r-\sum_{r=1}^{n} u_r$$ to later give us the required working to solution... ... ##4n^2(4n+3)-n^2(2n+3)=16n^3+12n^2-2n^3-3n^2=14n^3+9n^2## as indicated. My question is on...

Just earlier today i was practicing solving some ODEs with the power series method and when i did it to the infinite square well i noticed that my final answer for ##\psi(x)## wouldn't give me the quantised energies. My solution was $$\psi(x) = \sum^{\infty}_{n=0} k^{2n}(\cos(x) + \sin(x))$$...

Using the given rule for the ##x_n##, write $$ \sum_l y_l = x_1 + \frac{1}{2} x_2 + \frac{1}{6} x_2 + \frac{1}{3} x_3 + \frac{1}{12} x_2 + \frac{1}{12} x_3 + \frac{1}{20} x_2 + \cdots + \frac{1}{n} x_n $$ $$ = x_1 + \sum_{n=2}^\infty \frac{1}{n(n-1)} x_2 + \sum_{n=3}^\infty \frac{2}{n(n-1)} x_3...

this is the circuit this is the theoretical graph

Write and test the fib function in two linked files (Fib.asm, fib_main.asm). Your solution must be made up of a function called fib(N, &array) to store the first N elements of the Fibonacci sequence into an array in memory. The value N is passed in $a0, and the address of the array is passed in...

While working on a probability problem I accidentally found this relationship: $$\frac a b = \frac a {(b-1)} - \frac a {{(b-1)}^2} + \frac a {{(b-1)}^3} - \frac a {{(b-1)}^4} + ~...$$ I have done a bit of work on it myself, and have tried to research similar series. It seems to lead to some...

How to prove that \[ \sum_{i=1}^{\infty}\frac{1}{2^{3i}}\left(\csc^{2}\left(\frac{\pi x}{2^{i}}\right)+1\right)\sec^{2}\left(\frac{\pi x}{2^{i}}\right)\sin^{2}\left(\pi x\right)=1 \] for all \( x\in\mathbb{R} \). Using graph, we can see that the value of this series is 1 for all values of x...

Homework Statement:: Tell me if a sequence or series diverges or converges Relevant Equations:: Geometric series, Telescoping series, Sequences. If I have a sequence equation can I tell if it converges or diverges by taking its limit or plugging in numbers to see what it goes too? Also if I...

V1 = Q/30 In parallel: C = C1 + C2 V2 = Q/(60 + 90) V1/V2 = 3 V1 = 3V2

I have to find 2 solutions of this Bessel's function using a power series. ##x^2 d^2y/dx^2 + x dy/dx+ (x^2 -9/4)y = 0## I'm using Frobenius method. What I did so far I put the function in the standard form and we have a singularity at x=0. Then using ##y(x) = (x-x_0)^p \sum(a_n)(x-x_0)^n##...

Consider the series below; From my own calculations, i noted that this series can also be written as ##S_n##=##\dfrac {3}{8}##⋅##\dfrac {4^n}{3^n}##. If indeed that is the case then how do we find the limit of my series to realize the required solution of ##1## as indicated on the textbook? I...

Greetings! I have a question about one assumption regarding this question even though I agree with the answer but I have a doubt about A, because when we study the convergence of a serie we use the assymptotic approximation, so why A is not correct? thank you! when we

Greetings! Here is the solution that I understand very well I reach a point I think the Professor has mad a mistake , which I need to confirm after putting x-1=t we found: But in this line I think there is error of factorization because we still need and (-1)^(n+1) over 3^n Thank you...

Greetings! I have a problem with the solution of that exercice I don´t agree with it because if i choose to factorise with 6^n instead of 2^n will get 5/6 instead thank you!

The only thing I could think of was using Q=sqrt(6000/50)-1 = 10.9, which then gets me XL=545 and XC=550, or 96.4nH, and 0.32pF at resonance of 900Mhz. I tried seeing if Zin=Zout equation would bring me close, so I tried Z=545+(50-550)=45, then XL for 38.5nH was 217.7, and XC for 2.4pF was 74...

This is the question, its long since i studied convergence...I need to attempt all the questions (attached)i will therefore make an attempt to answer one part at a time i.e ##a## first. wawawawawa! does not look nice...let me look at my old notes on this chapter then i will respond...

First series \frac{1}{2}\sum^{\infty}_{n=0}\frac{(-1)^n}{n+1}(\frac{1}{p^2})^{n+1}= \frac{1}{2}(\frac{1}{p^2}-\frac{1}{2p^4}+\frac{1}{3p^6}-\frac{1}{4p^8}+...) whereas second one is...

Summary:: How to know which one is bigger when n goes to infinity? $$ \sum_{n=1}^\infty \frac {1} {\sqrt {n}(\sqrt {n+1}+\sqrt {n-1})} $$ And: $$ \sum_{n=1}^\infty \frac {1} {\sqrt {n}(\sqrt {n}+\sqrt {n})} $$ I thought at first that the second one is bigger, although, I came to realize, to my...

(a) i tried to decompose the fracion as a sum of fractions of form ##\frac{1}{1-g}## $$f=\frac{-z}{(1+z)(2-z)}=\frac{a}{1+z}+\frac{b}{2-z}$$ $$a=\frac{1}{3}, b=-\frac{2}{3}$$ $$f=\frac{1}{6}\frac{1}{1+z}-\frac{1}{3}\frac{1}{1-\frac{z}{2}}$$ $$f=\frac{1}{6}\sum_{n=0}^\infty...

If light at a known polarity goes through a beam splitting polarizer and then goes through the reverse orientation of that polarizer it will exit with the same polarization that it entered with. What happens if you put light through a series of beam splitting polarizers and then through the...

The following is my attempt at the solution. Here, I used limit comparison test to arrive at the answer that the series converges. However, the answer sheet reads that the series diverges. I am confused because I cannot figure where my work went wrong… can anyone tell me how the series...

The answer sheet states that the series converges by limit comparison test (the second way). In the case of this particular problem, would it be also okay to use the comparison test, as shown above? (The first way) Thank you!

I need to develop $\mathrm{ln}(x)$ into series, where $x \geq 1$, and I don`t know how? In literature I only found series of $\mathrm{ln}(x)$, where: 1. $|x-1| \leq 1 \land x \neq 0$, $ \,\,\,\,\, \mathrm{ln}(x) = x - 1 - \dfrac{(x-1)^2}{2} + ...$ 2. $|x| \leq 1 \land x \neq -1$, $ \,\,\,\,\...

I teach electricity in grade 9. For the concept of conductors, they are described in the textbook as atoms where the outer electrons can easily move from one atom to another (e.g. copper). But I noticed that on the triboelectric series, copper and other metals are listed as having a strong(er)...

From the statement above, since the ring is massless, there's no force acting vertically on the rings. Thus, the slope is null. ##\frac{\partial y(0,0)}{\partial x} = \frac{\partial y(L,0)}{\partial x} = 0## ##\frac{\partial y(0,0)}{\partial x} = A\frac{2 \pi}{L}cos(\frac{2 \pi 0}{L}) =...

Is it valid to use limit comparison test to compute the following series? If it is, would my reasoning be valid? Thank you!