Series Definition and 998 Threads

While transforming the equation of the Basel problem, the following infinite series was obtained. $$\sum_{n=1}^{\infty} \frac{n^2+3n+1}{n^4+2n^3+n^2}=2$$ However I couldn't think of a simple way to prove that. Can anyone prove that this equation holds true?

∞ ∑ (n∧2+3n+1) / (n∧4+2n∧3+n∧2) =? n=1 I attempted to find the general sum of this 'expression'?? But no luck. How can I solve this?

Hi. Any fans of DARK here? This show was mind blowing. The feel of the show may be to heavy for some, especially in the beginning. So it may not be your style. But if you like that atmosphere, the series is truly amazing. The writing is on a whole new level. If you haven't watched it, I...

Dear Everyone, I am wondering how to use the integral formula for a holomorphic function at all points except a point that does not exist in function's analyticity. For instance, Let $f$ be defined as $$f(z)=\frac{z}{e^z-i}$$. $f$ is holomorphic everywhere except for $z_n=i\pi/2+2ni\pi$ for...

I just watched season one of Tales from the Loop. It has the mood of Interstellar on the Earth where people live quiet lives of desperation. There’s an underground physics lab nicknamed the Loop where the impossible becomes possible. There’s the people whose lives are affected in strange ways...

##\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##?

First, I try to define the function in the figure above: ##V(t)=100\left[sin(120\{pi}\right]##. Then, I use the fact that absolute value function is an even function, so only Fourier series only contain cosine terms. In other words, ##b_n = 0## Next, I want to determine Fourier coefficient...

Hello Everyone! I want to learn about Fourier series (not Fourier transform), that is approximating a continuous periodic function with something like this ##a_0 \sum_{n=1}^{\infty} (a_n \cos nt + b_n \sin nt)##. I tried some videos and lecture notes that I could find with a google search but...

I tried to use the ratio test, but I am stuck on finding the range of the limit. $$\because \left|x-1\right|<1.5=Radius$$ $$\therefore -0.5<x<2.5$$ $$\lim _{n \to \infty} \left| \frac{A_{n+1}(x-1)^{n+1}}{A_n(x-1)^n} \right|$$ $$\lim_{n \to \infty} \frac{A_{n+1} \left|x-1\right|}{A_n} <1$$...

Hi, I was watching a video on the origin of Taylor Series shown at the bottom. Question 1: The following screenshot was taken at 2:06. The following is said between 01:56 - 02:05: Halley gives these two sets of equations for finding nth roots which we can generalize coming up with one...

Hi, A person has 40 litres of milk. As soon as he sells half a litre, he mixes the remainder with half a litre of water. How often can he repeat the process, before the amount of milk in the mixture is 50% of the whole? Detailed explanation is appreciated.:) Solution: I am working on...

a) I think I got this one (I have to thank samalkhaiat and PeroK for helping me with the training in LTs :) ) $$\eta_{\mu\nu}\Big(\delta^{\mu}_{\rho} + \epsilon^{\mu}_{ \ \ \rho} +\frac{1}{2!} \epsilon^{\mu}_{ \ \ \lambda}\epsilon^{\lambda}_{ \ \ \rho}+ \ ...\Big)\Big(\delta^{\nu}_{\sigma} +...

Here is a circuit diagram: . We have three capacitors, with capacitances ##C_1##, ##C_2## and ##C_3##. Plates are labelled as ##A_1, A_2, A_3 ... A_6##. Point P is connected to the positive terminal of the battery and point N is connected to the negative terminal of the...

Known: V source = 30.0 V , R1 = 15.0 W, R2 = 15.0 W, R3 = 15.0 W To determine the current, first find the equivalent resistance. I = Vsource/R and R = RA + RB = Vsource/RA + RB 30.0 V/15.0 W + 15.0 W + 15.0 W = 1.5 A This is as far as I could do the work for this question. I’m having trouble..

Hi All, I've been going through Shankar's 'Principles of Quantum Mechanics' and I don't quite understand the point the author is trying to make in this exercise. I get that this wavefunction is not a solution to the Schrodinger equation as it is not continuous at the boundaries and neither is...

I recently stumbled upon Gradshteyn, Ryzhik: Table of Integrals, Series, and Products and it is worth recommending for all who have to deal with actual solutions, i.e. especially engineers, physicists and all who are confronted with calculating integrals, series and products.

Summary:: If ##f(x)=-f(x+L/2)##, where L is the period of the periodic function ##f(x)##, then the coefficient of the even term of its Fourier series is zero. Hint: we can use the shifting property of the Fourier transform. So here's my attempt to this problem so far...

Mentor note: Moved from technical section, so is missing the homework template. Im doing some older exams that my professor has provided, but I haven't got the solutions for these. Can someone help confirm that the solutions I've arrived at are correct?

Modern batteries use double-sided anode and cathodes for greater energy density. Series wiring of batteries is typically accomplished by connecting the anode of one cell to the cathode of another. However, can series be accomplished by stacking double-sided anode and cathode alternatingly with...

What is the last digit of $1+2+\cdots+n$ if the last digit of $1^3+2^3+\cdots+n^3$ is 1?

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?

Hello. I have completed the following question. My answer: i) Circuit Impedance Reactance = XL = 2 x pi x F x L = 2 x pi x 50 x 0.15 = 47.12 Ohms Reactance of Capacitor = XC = 1/2 x pi x F x C...

Dear Everyone, I am having trouble with finding a formula of the multiplication 3 formula power series. \[ \left(\sum_{n=0}^{\infty} a_nx^n \right)\left(\sum_{k=0}^{\infty} b_kx^k \right)\left(\sum_{m=0}^{\infty} c_mx^m \right) \] Work: For the constant term: $a_0b_0c_0$ For The linear...

Set ##\epsilon=\frac{1}{2}##. Let ##N\in \mathbb{N}## and choose ##n=N,m=2N##. Then: ##\begin{align*} \left|s_N-s_{2N}\right|&=&\left|\sum_{l=1}^N \frac{1}{l} - \sum_{l=1}^{2N} \frac{1}{l}\right|\\...

Hi all, In an LRC AC series circuit, at which frequencies are where you are mainly dumping your generator/current energy into capacitor to create electric fields or into the inductor to create magnetic fields? So, for example, at low frequencies, f --> 0, the impedance of the inductor goes to 0...

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!

Not really a homework problem, just an equation from my textbook that I do not understand. I can't think of any way to even begin manipulating the right hand side to make it equal the left hand side. Just to confirm equality (thanks to another user for suggestion), I multiplied both sides by of...

LIGO India EPO (Education and Public Outreach) team is hosting a series of talks on Youtube. No registration or any formalities; just tune into the LIGO India EPO Youtube channel and you can attend the lectures. Following is the list of upcoming talks: 20th April: Speaker: Prof. Ajith...

Interestingly, If I neglect the ##(-1)^n## or ##(-1)^{n+1}## then apply preliminary test, I could find the limit. Whether the limit is not equal to zero, as in series number 1 and 2, then I can conclude the series is divergent. But, if the limit is equal to zero, as in series number 3, then I...

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.

I have used ##\sim## but meant ##\sum_{k=0}^\infty## my math homework platform is telling me that this is wrong. I've tried using desmos to test it and it was a perfect match. Any ideas on where I went wrong?

This is a second grade high school problem, translated from my native language. I don't have a problem with calculating, but with understanding the concept. There is an instruction with the assignment that says: The capacitor can be viewed as a combination of two capacitors in series with...

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I am revising perturbation theory from Griffiths introduction to quantum mechanics. Griffiths uses power series to represent the perturbation in the system due to small change in the Hamiltonian. But I see no justification for it! Other than the fact that it works. I searched on the internet a...

Can anyone help with this problem. I've tried integral test but seems to be too complicated.

https://scholar.harvard.edu/files/david-morin/files/relativity_chap_1.pdf The questions start at page 44 Whenever I refer to y, y = gamma. 1.1 This question is primarily deriving LV/C^2? How does 2LV / c^2-v^2 becomes 2Lv / c^2(1-v^2/c^2)1.4 On the solution page it shows fig 1.61 and fig...

I found total capacitance and inserted the total capacitance and emf of cell in equation CV =Q. However I know that there is a resistor connected so that this accounts for lost volts

I'm currently typing up some notes on topics since I have free time right now, and this question popped into my head. Given a problem as follows: Find the first five terms of the Taylor series about some ##x_0## and describe the largest interval containing ##x_0## in which they are analytic...

I know that the Lyman series is a series of lines in the ultra-violet. So that means a higher frequency so it will fall on the right of the diagram. And Paschen is infrared. So a lower frequency so on the left side of the diagram?

A Harvard teams believe they have found a protein series they call "Hemolithin" in an asteroid. Isotopes and other evidence indicates that it is not from a terrestrial source."Astrobiology Web" link arxiv pdf link

We transform the series into a power series by a change of variable: y = √(x2+1) We have the following after substituting: ∑(2nyn/(3n+n3)) We use the ratio test: ρn = |(2n+1yn+1/(3n+1+(n+1)3)/(2nyn/(3n+n3)| = |(3n+n3)2y/(3n+1+(n+1)3)| ρ = |(3∞+∞3)2y/(3∞+1+(∞+1)3)| = |2y| |2y| < 1 |y| = 1/2...

i tired using complex identity equation for sin(pi*k/3) but it doesn't work out

∑((√(x2+1))n22/(3n+n3)) We use the ratio test: ρn = |2(3n+n3)√(x2+1)/(3n+1+(n+1)3)| ρ = |2√(x2+1)| ρ < 1 |2√(x2+1)| < 1 No "x" satisfies this expression, so I conclude the series doesn't converge for any "x". However the answer in the book says the series converges for |x| < √(5)/2. What am...

The series ##\sum_{n=0}^\infty \left( ne^{\frac 3 n}-n \right) \left ( \sin \frac {\alpha} {n} - \frac 5 n\right)## i did ##\sum_{n=0}^\infty n\left( e^{\frac 3 n}-1 \right) \left ( \sin \frac {\alpha} {n} - \frac 5 n\right)## for n going to infinity ## \left( e^{\frac 3 n}-1 \right)##...

∑(x2n/(2nn2)) We use the ratio test: ρn = |(x2n2/(2(n+1)2)| ρ = |x2/2| ρ < 1 |x2| < 2 |x| = √(2) We investigate the endpoints: x = 2: ∑(4n/(2nn2) = ∑(2n/n2)) We use the preliminary test: limn→∞ 2n/n2 = ∞ Since the numerator is greater than the denominator. Therefore, x = 2 shouldn't be...

My partner asked me about questions no. 8 and 9. Number 8 asks about what is the area of the quadrilateral. Number 9 asks about what number is below the number 25. Those are questions for Elementary School Math Olympiads in my country but both of us were having a hard time figuring them out...

Moved from technical forum section, so missing the homework template Let x be a real number. Find the function whose power series is represented as follows: x^3/3! + x^9/9! + x^15/15! ... I see that there is a connection to the power series expansion of e^x but am having difficulty finding...

Suppose I have a galvanic cell, where I've arbitrarily set the (-) anode to have a potential of zero volts and the (+) cathode to ##\epsilon## V. The electrodes are connected via the load, but also via the solutions and salt bridge in the centre. Edit: The two trailing wires are connected to a...

At the exam i had this power series but couldn't solve it ##\sum_{k=0}^\infty (-1)^\left(k+1\right) \frac {k} {log(k+1)} (2x-1)^k## i did apply the ratio test (lets put aside for the moment (2x-1)^k ) to the series ##\sum_{k=0}^\infty \frac {k} {log(k+1)}## in order to see to what this...

Summary:: Trying to find Rth but I do not get the same value as the one from the solution. [moderator: moved from a technical forum. No template.] I am trying to find Rth to solve this problem, however once I simplified it, I get a value of 700.745 Ω while in the solution, the answer is...