Series Definition and 998 Threads

Hello there. I'm here to request help with mathematics in respect to a problem of quantum physics. Consider the following function $$ f(\theta) = \sum_{l=0}^{\infty}(2l+1)a_l P_l(cos\theta) , $$ where ##f(\theta)## is a complex function ##P_l(cos\theta)## is the l-th Legendre polynomial and...

Hi I have a series ${f}_{1}$ , ${f}_{2}$, ... that are all a functions of a variable $t$ I am seeking a point-wise convergence. to investigate the convergence of the series I took Laplace transform. If I can find a condition on the Laplace variable $s$, can I find the condition of convergence...

1. ... Expand the Eigenvalue as a power series in epsilon, up to second order: λ=[3+√(1+4 ε^2)]V0 / 2 Homework Equations I am familiar with power series, but I don't know how to expand this as one.[/B]The Attempt at a Solution :[/B] I have played around with the idea of using known power...

Homework Statement Homework Equations All Fourier series trigonometric equations. I think we are required to use sigma function, integrals, etc.[/B]The Attempt at a Solution We are currently working through our Fourier series revision studying integrals of periodic functions within K.A...

Q: Suppose ##u(x,t)## satisfies the heat equation for ##0<x<a## with the usual initial condition ##u(x,0)=f(x)##, and the temperature given to be a non-zero constant C on the surfaces ##x=0## and ##x=a##. We have BCs ##u(0,t) = u(a,t) = C.## Our standard method for finding u doesn't work here...

I was given a function that is periodic about 2π and I need to plot it. I was wondering if there is a way to input a value and have mathematica generate a new graph with the number of iterations. The function is: $$\sum_{n=1}^{N}\frac{sin(nx)}{n}$$ where n is an odd integer. I guess a better...

Homework Statement Fourier series expansion of a signal f(t) is given as f(t) = summation (n = -inf to n = +inf) [3/(4+(3n pi)2) ) * e j pi n t A term in expansion is A0cos(6 pi ) find the value of A0 Repeat above question for A0 sin (6 pi t) Homework Equations Fourier expansion is summation...

Homework Statement For example cosh(x) = 1+x2/2!+x4/4!+x6/6!+... Homework EquationsThe Attempt at a Solution So plugging in x=0 you get that coshx = 1 at the origin. The approximate graph for the coshx function up to the second order looks like a 1+x2/2! graph, but what about graphing coshx...

Homework Statement The problem from the textbook is: Is it possible to connect resistors together in a way that cannot be reduced to some combination of series and parallel combinations? Homework Equations V = IR kirchhoff's current law kirchhoff's loop law The Attempt at a Solution I am...

Say you have two separate resistors in a circuit. When you close the switch on the circuit is it the electric field that flows through the circuit that effectively sets the voltage drops so a bigger voltage drop occurs across the higher resistor in proportion to its resistance such that the...

Homework Statement Use a comparison test to determine whether this series converges: \sum_{x=1}^{\infty }\sin ^2(\frac{1}{x}) Homework EquationsThe Attempt at a Solution At small values of x: \sin x\approx x a_{x}=\sin \frac{1}{x} b_{x}=\frac{1}{x} \lim...

Homework Statement \sum_{x=2}^{\infty } \frac{1}{(lnx)^9} Homework EquationsThe Attempt at a Solution x \geqslant 2 0 \leqslant lnx < x 0 < \frac{1}{x} < \frac{1}{lnx} From this we know that 1 / lnx diverges and I wanted to use this fact to show that 1 / [(lnx) ^ 9] diverges but at k...

Homework Statement Find the Fourier series of the function f(x) =√(x2) -pi/2<x<pi/2 , with period pi Homework EquationsThe Attempt at a Solution I have tried attempting the question, but couldn't get the answer. uploaded my...

Homework Statement Find the power series in x-x0 for the general solution of y"-y=0; x0=3. Homework Equations None. The Attempt at a Solution Let me post my whole work:

Assuming the resistance of a wire in a series circuit, consisting only of 1 component (e.g. filament lamp) and a battery, is negligible; does each Coulomb of charge commit all of its electrical potential energy, supplied by the battery's potential difference, as work done across the component...

Homework Statement No actual problem, thinking about the telescoping series theorem and Grandi's series For reference Grandi's series S = 1 - 1 + 1 - 1... Homework Equations [/B] The telescoping series theorem in my book states that a telescoping series of the form (b1 - b2) + ... + (bn -...

Self Study 1. Homework Statement Consider a periodic function f (x), with periodicity 2π, Homework Equations ##A_{0} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)dx## ##A_{n} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)cos\frac{2\pi rx}{L}dx## ##B_{n} =...

I'm studying QFT in the path integral formalism, and got stuck in deriving the Schwinger Dyson equation for a real free scalar field, L=½(∂φ)^2 - m^2 φ^2 in the equation, S[φ]=∫ d4x L[φ] ∫ Dφ e^{i S[φ]} φ(x1) φ(x2) = ∫ Dφ e^{i S[φ']} φ'(x1) φ'(x2) Particularly, it is in the Taylor series...

Homework Statement Using the taylor series at point ##(x=0)## also known as the meclaurin series find the limit of the expression: $$L=\lim_{x \rightarrow 0} \frac{1}{x}\left(\frac{1}{x}-\frac{cosx}{sinx}\right)$$ Homework Equations 3. The Attempt at a Solution [/B] ##L=\lim_{x \rightarrow 0}...

Homework Statement Hello, I am currently doing some holiday pre-study for signals analysis coming up next semester. I'm mainly using MIT OCW 6.003 from 2011 with some other web resources (youtube, etc). The initial stuff is heavy on the old infinite series stuff, that seems often skimmed...

I have in my lecture notes that ##E_{k}(t=0) =1 ##, ##E_k (t)## the Eisenstein series given by: ##E_k (t) = 1 - \frac{2k}{B_k} \sum\limits^{\infty}_1 \sigma_{k-1}(n) q^{n} ## ##B_k## Bermouli number ##q^n = e^{ 2 \pi i n t} ## context modular forms. Also have set ##lim t \to i\infty = 0## ...

So I am trying to figure out how much Wh I would have with these solar cells I have. Each solar cell is rated to have 2.8w. Does this mean if I have 40 of them I would have 112 wh? I am going to be putting the solar cells in series. Does this affect the power? I know adding in series is good...

Homework Statement Find the values of the constant a for which the problem y''(t)+ay(t)=y(t+π), t∈ℝ, has a solution with period 2π which is not identically zero. Also determine all such solutions Homework Equations With help of Fourier series I know that : Cn(y''(t))= -n2*Cn(y(t)) Cn(y(t+π)) =...

Hi I am supposed to find solution of $$xy''+y'+xy=0$$ but i am left with reversing this equation. i am studying solution of a differential equation by series now and I cannot reverse a series in the form of: $$ J(x)=1-\frac{1}{x^2} +\frac{3x^4}{32} - \frac{5x^6}{576} ...$$ $$...

Homework Statement Find the sum of the following series: $$ \left( \frac 1 2 + \frac 1 4 \right) + \left( \frac 1 {2^2} + \frac 1 {4^2} \right) +~...~+ \left( \frac 1 {2^k} + \frac 1 {4^k} \right) +~...$$ Homework Equations $$ \sum_{n = 1}^{\infty} \left( u_k+v_k \right) = \sum_{n =...

Svein submitted a new PF Insights post Using the Fourier Series To Find Some Interesting Sums Continue reading the Original PF Insights Post.

[Please excuse the screengrabs of the fomulae - I'll get around to learning TeX someday!] 1. Homework Statement Find the sum of this series (answer included - not the one I'm getting) The Attempt at a Solution So I'm trying to sum this series as a telescoping sum. I decomposed the fraction...

[mentor note: thread moved from non-hw forum to here hence no homework template] Can someone explain to me how it is that $$\sum_{n=a}^b (2n+1)=(b+1)^2-a^2$$ I thought it would be $$\sum_{n=a}^b (2n+1)=(2a+1)+(2b+1)$$ but I am clearly very wrong. I would greatly appreciate any help.

Homework Statement A fishery manager knows that her fish population naturally increases at a rate of 1.4% per month, while 119 fish are harvested each month. Let Fn be the fish population after the nth month, where F0 = 4500 fish. Assume that that process continues indefinitely. Use the...

Homework Statement I have been working through “Basic Concepts in Relativity and Early Quantum Theory” by Resnick and Halliday. I've read about and done most of the problems about time-dilation, length-contraction, and Doppler effect. But then I got to problem 2-76, and I’ve been swirling in...

<OP warned about not using the homework template> Obtain a series solution of the differential equation x(x − 1)y" + [5x − 1]y' + 4y = 0Do I start by solving it normally then getting a series for the solution or assume y=power series differentiate then add up the series? I did the latter and...

< Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown >[/color] Hi all. I'm completely new to these forums so sorry if I'm doing anything wrong. Anyway, I have this question... Find the Fourier series for the periodic function f(x) = x^2 (-pi < x < pi)...

Hi, I'm starting to studying Fourier series and I have troubles with one exercises of complex Fourier series with f(t) = t: $$t=\sum_{n=-\infty }^{\infty } \frac{e^{itn}}{2\pi }\int_{-\pi}^{\pi}t\: e^{-itn} dt$$ $$t=\sum_{n=-\infty }^{\infty } \frac{cos(tn)+i\, sin(tn)}{2\pi...

Hi. I helped my neighbour putting up (quite old, no LEDs) strings of Christmas lights and noted that some of them (different brands) are connected in parallel, others in series. Inevitably, we found several of the serial strings not working due to defective bulbs. We replaced the visibly...

Suppose that the Taylor series of a function ##f: (a,b) \subset \mathbb{R} \to \mathbb{R}## (with ##f \in C^{\infty}##), centered in a point ##x_0 \in (a,b)## converges to ##f(x)## ##\forall x \in (x_0-r, x_0+r)## with ##r >0##. That is $$f(x)=\sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n...

$\tiny{206.r2.11}$ $\textsf{find the power series represntation for $\displaystyle f(x)=\frac{x^7}{3+5x^2}$ (state the interval of convergence), then find the derivative of the series}$ \begin{align} f(x)&=\frac{x^7}{3}\implies\frac{1}{1-\left(-\frac{5}{3}x^2\right)}&(1)\\...

can anyone refer me to a graphical representation of how this works so i can build one?

We made a RLC circuit in the lab and took some values of R and LC Voltage while we changed the frequency. So the experimental data seem to suggest that at resonance (VLC=18 mV : min) the Voltage of the resistor is 834 mV. But the initial voltage given was measured 1.426 V (All Values rms)...

Homework Statement and the solution (just to check my work) Homework Equations None specifically. There seems to be many ways to solve these problems, but the one used in class seemed to be partial fractions and Taylor series. The Attempt at a Solution The first step seems to be expanding...

If we can use Regression analysis to forecast, why do we use “Time Series Decomposition”? What's the difference between the 2? Thanks

I have this: $$ \sum_{n = 1}^{\infty} \frac{n^n}{3^{1 + 3n}}$$ And I need to determine if it is convergent or divergent. I try the limit comparison test against: $$ \frac{1}{3^{1 + 3n}}$$. So I need to determine $$ \lim_{{n}\to{\infty}} \frac{3^{1 + 3n} \cdot n^n}{3^{1 + 3n}}$$ Or $$...

$\textsf{a. Find the first four nozero terms of the Maciaurin series for the given function} \\$ \begin{align} f^0(x)&=\ln{ (6 x + 1)} &\therefore f^0(a)&=0\\ f^1(x)&=\frac{6}{(6 x + 1)} &\therefore f^1(a)&=6\\ f^2(x)&= \frac{-36}{(6 x + 1)^2} &\therefore f^2(a)&=-36\\ f^3(x)&= \frac{432}{(6 x...

I have $$\sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$$ I'm trying the limit comparison test, so I let $$ b = \frac{1}{n^{\frac{9}{8}}}$$ and $a = \sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$ $\frac{a}{b} = (lnn)^ {12}$ therefore I know the limit of this as n...

Homework Statement consider ##f## a meromorphic function with a finite pole at ##z=a## of order ##m##. Thus ##f(z)## has a laurent expansion: ##f(z)=\sum\limits_{n=-m}^{\infty} a_{n} (z-a)^{n} ## I want to show that ##f'(z)'/f(z)= \frac{m}{z-a} + holomorphic function ## And so where a...

$\textsf{a. Find the first three nonzero terms of the Taylor series $a=\frac{3\pi}{4}$}$ \begin{align} \displaystyle f^0(x)&=\sin{x} &\therefore \ \ f^0(a)&=\sin{x} \\ f^1(x)&=\cos{x} &\therefore \ \ f^1(a)&= -\frac{\sqrt{2}}{2}\\ f^2(x)&=- \sin{x}&\therefore \ \ f^2(a)&=\frac{\sqrt{2}}{2} \\...

multiple 220V to 5v, 2A ac to dc adapters connected to the same 2-phase input terminals input voltage is 220v domestic supply. is it safe to i join output in series to obtain 10v /15v/25v etc. ?

$\textsf{a. Find the first four nonzero terms of the Taylor series $a=1$}$ \begin{align} \displaystyle f^0(x)&=6^{x} &\therefore \ \ f^0(a)&= 6 \\ f^1(x)&=6^{x}\ln(6) &\therefore \ \ f^1(a)&= 6\ln(6) \\ f^2(x)&={6^{x}\ln(6)^2} &\therefore \ \ f^2(a)&= {12\ln(6)} \\ f^3(x)&={6^{x}\ln(6)^3}...

$\tiny{206.11.3.39}$ $\textsf{a. Find the first four nonzero terms of the Taylor series $a=0$}$ \begin{align} \displaystyle f^0(x)&=(1+x)^{-2} &\therefore \ \ f^0(a)&= 1 \\ f^1(x)&=\frac{-2}{(x+1)^3} &\therefore \ \ f^1(a)&= -2 \\ f^2(x)&=\frac{6}{(x+1)^4} &\therefore \ \ f^2(a)&= 6 \\...

$\textsf{a. Find the first four nozero terms of the Maciaurin series for the given function} \\$ \begin{align} a&=0 \\ f(x)&=(-5+x^2)^{-1} \\ \\ f^0(x)&=(-5+x^2)^{-1}\therefore f^0(a) = 1 \\ f^1(x)&=\frac{-2x}{(x^2-5)^2} \therefore f^1(a) = 0 \\ f^2(x)&=\frac{2(3x^3+5)}{(x^2-5)^3} \therefore...

Is the series of numbers 2,3,5,8,13,21 ... a fibronacci sequence ? Because it doesn't start with 1 , but it fulfills the explicit formula .