Series Definition and 998 Threads

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I am reading an interesting book by Julian Havil called:" Gamma-Exploring Euler's Constant." Much of the book is devoted to the harmonic series,a slowly diverging series that tends toward infinity.However,one paragraph puzzles me. On p. 23 he says: " In 1968 John W. Wrench Jr calculated the...

I think I have mistaken something here but does changes in voltage affect the frequency of a series LC or parallel LC circuit or not? Or is the frequency only dependent on capacitance and inductance of the circuit elements? And if the fed in frequency matches the resonant frequency then the LC...

One property of series resonance circuit is that at resonance, the voltage across circuit elements R,L and C may be larger than the source voltage. I can relate it to vector analogy where component vectors may have larger values than the resultant and the phenomenon is counter-intuitive. This...

Firstly, the matrix notation of the series is, \begin{align*} f\left(x, y, z\right) &= f\left(a, b, c\right) + \left(\mathbf{x} - \mathbf{a}\right)^T \frac{\partial f\left(a, b, c\right)}{\partial \mathbf{x}} + \frac{1}{2}\left(\mathbf{x} - \mathbf{a}\right)^T \frac{\partial^2 f\left(a, b...

Evaluate ##\lim_{n \rightarrow +\infty} \frac {1} {n} [(\frac {1}{n})^{1.5} + (\frac {2}{n})^{1.5} +(\frac {3}{n})^{1.5}+ (\frac {4}{n})^{1.5}+...+(\frac {n}{n})^{1.5}]## Hello. So I'm solving this question at the moment. I know I'm supposed to find out the summation of this before being able...

So I'm on page 67 of Marion/Thornton's "Classical Dynamics of Particles and Systems" and I'm in need of some help. I understand that so far there's is an equation that cannot be solved analytically (regarding motion due to the air resistance and finding the range of the an object shot from a...

Since there are many Star Wars and SF fans on PF, I wanted to share some info about the upcoming Star Wars tv series The Mandalorian, scheduled to premiere November 12, 2019. Minor spoilers below (info about setting and background): As far as I know there has not been any teaser or trailer...

I am trying to make sense of the wikipedia article section regarding Cauchy product of several series. but am stuck right at the start because the notation used there is unfamiliar to me and not explained previously in the article. The commas in ##\Sigma a_1, k_1## etc. mean nothing to me. Am I...

This attractor is unusual because it uses both the tanh() and abs() functions. A picture can be found here (penultimate image). Here is some dependency-free Python (abridged from the GitHub code, but not flattened!) to generate the data to arbitrary order: #!/usr/bin/env python3 from sys...

I tried diffrentiating upto certain higher orders but didn’t find any way.. is there a trick or a transformation involved to make this task less hectic? Pls help

Maybe introduce a parametric factor can be help.

Hello, so for a Fourier series in the interval [-L,L] with L=L and T=2L the coefficients are given by $$a_0=\frac{1}{L}\int_{-L}^Lf(t)dt$$ $$a_n=\frac{1}{L}\int_{-L}^Lf(t)\cos{\frac{n\pi t}{L}}dt$$ $$b_n=\frac{1}{L}\int_{-L}^Lf(t)\sin{\frac{n\pi t}{L}}dt$$ But if we have an interval like [0,L]...

Here is the actual question. And here is my attempt at a solution In Summary I did the following Found the Equivalence Resistance to Be 5.9 ohms and the Current throughout the entire resistor to be 1.53 Amperes Worked backwards from my resistor simplifications. When the resistors were in...

Hi, I would like to know how to calculate the total capacitance for a capacitor that has a certain plate overlap area and two dielectrics in between, one being a solid state dielectric and the other one being air. This is not a school project, I just thought about it and tried to calculate...

Hi, I was trying to solve the following problem myself but couldn't figure out how the given Taylor series for log(x) is found. Taylor series for a function f(x) is given as follows. Question 1: I was trying to find the derivative of log(x). My calculator gives it as...

I first solved the first two terms and then i solved the resulting term with the third term and so on.At last i was left with x^n/((x-a1)(x-a2)...(x-an)) .Thrn i took log on both sides and then differentiated both sides with respect to x.I got 1/y dy/dx=n/x -1/(x-a1)-1/(x-a2)...-1/(x-an).But now...

Summary: Series RLC and Parallel RLC circuits How can the voltage across a capacitor or inductor in a series RLC circuit be greater than the applied AC source voltage? The formula suggest that either can be larger than the source voltage but I still find it counter intuitive. Also for...

I tried substituting x=cos2theta but it was of no use.I thought many ways but i could not make 0/0 form.So please help.

Hello, I'm trying to solve this, any idea please? Basically: Demonstrate for the next three processes if the Time Series would be stationary, if not, it should establish the conditions for it to be stationary. Thanks

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with the proof of Theorem 2.3.9 (a) ... Theorem 2.3.9 reads as follows: Now, we can prove Theorem 2.3.9 (a) using the Cauchy...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help In order to formulate a rigorous proof to the proposition stated in Exercise 2.3.10 (1) ... ... Exercise 2.3.10 (1) reads as...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with an aspect of the proof of Proposition 2.3.8 ... Proposition 2.3.8 and its proof read as follows: In the above proof by...

I think ##\lim_{n\rightarrow \infty} a_n = 0## since by direct substitution the value of limit won't be equal to 2 so by direct substitution we must get indeterminate form. Then how to check for ##\sum_{n=1}^\infty a_n##? I don't think divergence test, integral test, comparison test, limit...

Because the Taylor series centered at 0, it is same as Maclaurin series. My attempts: 1st attempt \begin{align} \frac{1}{1-x} = \sum_{n=0}^\infty x^n\\ \\ \frac{1}{x} = \frac{1}{1-(1-x)} = \sum_{n=0}^\infty (1-x)^n\\ \\ \frac{1}{x^2} = \sum_{n=0}^\infty (1-x^2)^n\\ \\ \frac{1}{(2-x)^2} =...

Hi, A function which could be represented using Fourier series should be periodic and bounded. I'd say that the function should also integrate to zero over its period ignoring the DC component. For many functions area from -π to 0 cancels out the area from 0 to π. For example, Fourier series...

I enjoyed a lot the three first volumes of Zeidler's planned series of 6 books on QFT. Unfortunately, he passed away too soon. However, it is clear from reading the first three books, that an outline of the next books in the series was already planned. Is there a draft, containing the basis of...

In the 1960s there was a series of science-themed novels for middle school children. Each book began with a boy accidentally meeting a scientist or science-club member and being introduced to the subject. I remember one about rockets, one about geology (and one about archeology?). I don't...

Does anyone watch the Chernobyl tv series? https://www.imdb.com/title/tt7366338/ It has such realism you would think it's right in the scene. Imdb rated it 9.6 out 10. Did all those details in the series actualy happened? Which part is dramatization? It made me think deep into the night...

The error ##e_{n}(y)## for ##\frac{1}{1-y}## is given by ##\frac{1}{(1-c)^{n+2}}y^{n+1}##. It follows that ##\frac{1}{1+y^2}=t_n(-y^2)+e_n(-y^2)## where ##t_n(y)## is the Taylor polynomial of ##\frac{1}{1-y}##. Taking the definite integral from 0 to ##x## on both sides yields that...

Hi. The #1 complaint about the Greiner series are the typos. It's a shame how it undermines the potential of this series which is, otherwise, highly acclaimed. Do you know where errata for any of these volumes can be found? Or perhaps those who have read some of the books could share their own...

## ## Well I start with equation 1): ## e^{b\theta }=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}e^{in\theta } ## If ## \theta =0 ## ##e^{b(0)}=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}e^{in(0) }## ##1=\frac{sinh(b\pi )}{\pi...

I'm not too sure how to use the hint here. What I had so far was this: an odd extension of ##f## implies ##f = \sum_{k=1}^\infty b_k \sin(k x)##. Notice for ##m>n## $$ \left|\sum_{k=1}^m b_k\sin(k x) - \sum_{k=1}^n b_k\sin(k x)\right| = \left| \sum_{k=n+1}^m b_k\sin(k x)\right| \leq...

The problem of the interaction of a point charge with a dielectric plate of finite thickness implies the existence of an infinite series of image charges (see http://www.lorentzcenter.nl/lc/web/2011/466/problems/2/Sometani00.pdf). I introduce notations identical to those used in this work. The...

Hi, I have some crucial questions belong to statistics: First, How can we derive the variance function with respect to mean for a given data? Secondly, I would like to ask: what method should we employ if the variance in time series behaves like a high order (such as ##𝑎𝑢_𝑡^5+𝑏𝑢_𝑡^4+𝑐𝑢_𝑡^3##...

For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.

Hi, as you know infinite sum of taylor series may not converge to its original function which means when we increase the degree of series then we may diverge more. Also you know taylor series is widely used for an approximation to vicinity of relevant point for any function. Let's think about a...

I've lately been interested in series and how they converge to interesting values. It's always interesting to see how they end up adding up to something involving pi or e or some other unexpected solution. I learned about the Leibniz formula back in college : pi/4 = 1/1-1/3+1/5-1/7+1/9-... and...

Hi, I've been reading the passage attached below and from what I understand we are looking at a 1D chain of atoms and if anyone atom moves it changes the potential for surrounding atoms and cause a change in energy in the system so the total energy is dependent on all the positions of the atoms...

I have already solved up to after the switches are flipped, and all the charge is on C1. See the second attached image for a detailed diagram of the situation after the switches are flipped. However, the notes then say that all the charge is trapped between C1 and C2, which I don't understand...

I am attempting to find the sine representation of cos 2x by letting $$f(x) = \cos2x, x>0$$ and $$-\cos2x, x<0$$ Which is an odd function. Hence using $$b_n = \dfrac{2}{l} \int^\pi _0 f(x) \sin(\dfrac{n\pi x}{l})dx$$ I obtain $$b_n = \dfrac{2n}{\pi} \left( \dfrac{(-1)^n - 1}{4-n^2} \right)$$...

Given that they're all on the same branch, I had assumed that they were in series with one another. But with the middle resistor having being on the middle of three branches, it looks parallel. Like I said, I have a feeling it's in series (making the answer 3R). This question is from a past...

f(x) = 4x/(x-3)^2 Find the first five non-zero terms of power series representation centered at x = 0. Also find the radius of convergence.

Hey all, it's been awhile since done any calculus or DE's but was trying out some modelling (best price price per item for bulk value deals as a function of time and amount), in the last line i have f(n,t) implicitly. Any pointers or techniques for solving such things?

A0 = 1 A1 = 3 3(An-1) / 4(An-2) = An

How do you increase torque in gerotor design other than increasing flow. Will lengthening it increase torque? Will increasing diameter increase torque? What about running 2 or 3 in series?

Curious about proving that ##\sum_{m=2}^\infty \sum_{n=2}^\infty 1/n^m ## = 1 ran this in Matlab and n,m to 2:1000 =0.9990, and n,m 2:10000 =0.9999, so it does appear to converge to 1

Homework Statement Hello, i am trying to do find the Fourier series of abs(sin(x)), but have some problems. As the function is even, bn = 0. I have calculated a0, and I am now working on calculating an. However, when looking at the solution manual, they have set up one calculation for n > 1...

Homework Statement I have encountered this problem from the book The Physics of Waves and in the end of chapter six, it asks me to prove the following identity as part of the operation to prove that as the limit of ##W## tends to infinity, the series becomes an integral. The series involved is...

Homework Statement Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n## Evaluate: ##f^{(8)}(4)## Homework Equations The Taylor Series Equation The Attempt at a Solution Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except for the...