Series Definition and 998 Threads

Homework Statement Fig. 25-39 shows a 12.0 V battery and four uncharged capacitors of capacitances C1 = 1.00 µF, C2 = 2.00 µF, C3 = 3.00 µF, and C4 = 4.00 µF. If only switch S1 is closed, what is the charge on (a) capacitor 1, (b) capacitor 2, (c) capacitor 3, and (d) capacitor 4? (figure at...

Homework Statement Find the Laurent expansions of ##f(z) = \frac{z+2}{z^2-z-2}## in ##1 < |z|<2## and then in ##2 < |z|< \infty## in powers of ##z## and ##1/z##. Homework Equations Theorem: Let ##f## be a rational function all of whose poles ##z_1,\dots , z_N## in the plane have order one and...

Homework Statement Find four terns of the Laurent series for the given function about ##z_0=0##. Also, give the residue of the function at the point. a) ##\frac{1}{e^z-1}## b) ##\frac{1}{1-\cos z}## Homework Equations The residue of the function at ##z_0## is coefficient before the...

I was experimenting with resonant frequency of a series RLC circuit: 5V AC source 10 ohms resistor 100microF capacitor 46mH inductor Resonant frequency is calculated to be around 74.2Hz. So I set the AC source to resonant frequency 74.2Hz and measured the voltage across the 10 ohms resistor...

Let $${a}_{n+1} = \frac{4}{7}{a}_{n} + \frac{3}{7}{a}_{n-1}$$ where a0 = 1, and a1 = 2. Find $$\lim_{{n}\to{\infty}}{a}_{n}$$ Well, seeing as it says that x approaches infinity, the difference between where points an-1, an, and an+1 are plotted on the y-axis is almost insignificant, so we can...

Is there any gamma decay or x-ray decay in Actinium series, Uranium series or Thorium series? On Wikipedia, it only shows alpha and beta decay, does it mean high energy photon decay (gamma or x-ray) exists in each process? Thank you! https://en.wikipedia.org/wiki/Decay_chain#Actinium_series

Homework Statement Find \lim_{x \to 0}\frac{ln(1+x^2)}{1-cos(x)} by using series representations. Check using L'Hospitals rule. Homework Equations Taylor polynomial at x=0: \sum_{k=0}^{\infty}\frac{f^{k}(0)}{k!}(x)^{k} = f(0) + f'(0)(x) + f''(0)x^{2} +... The Attempt at a Solution Using...

Hi. I am working on a linear algebra problem that arose somewhat like this: Suppose that you are shining a light with a known intensity spectrum P(\lambda) upon a surface with an unknown reflection spectrum, R(\lambda). You have a detector to detect the total reflected light intensity, I. How to...

Somewhere I saw that the sum of the infinite arithmetic series $$\sum_{n=1}^{\infty}n = \frac{-1}{12}$$ Why exactly is this? I thought infinite arithmetic series had no solution? Also... WHY is it negative? Seems counter-intuitive that the sum of all the NATURAL numbers is a decimal, a...

hey, sir i have a question why the current remain same in series combination of resistance?

Homework Statement I am doing #9. Homework EquationsThe Attempt at a Solution I've been looking at a lot of similar problems on the internet. The main difference between this one and them is that this one has an interval of [0,4] while they often have intervals of [0,pi] or [-pi,pi] In my...

Homework Statement Homework EquationsThe Attempt at a Solution So I am tasked with answer #3 and #4. I have supplied the indicated parenthesis of 8 also with the image. Here is my thinking: Take the Fourier series for |sin(θ)|. Let θ = 0 and we see a perfect relationship. sin(0) = 0 and...

what is the rationale of multiplying "r" to the second line of series? why does cancelling those terms give us a VALID, sound, logical answer? please help. here's a video of the procedure

Homework Statement Determine whether each of the following series is convergent or divergent. If the series is convergent, find its sum \sum_{i=1}^{\infty} \frac{6}{9i^{2}+6i-8} Homework Equations Partial fraction decomposition \frac{1}{3i-2} - \frac{1}{3i+4} The Attempt at a Solution...

Hello! (Wave) We have a sequence $(y_n)$ with $y_n \geq 0$. We assume that the series $\sum_{n=1}^{\infty} \frac{y_n}{1+y_n}$ converges. How can we show that the series $\sum_{n=1}^{\infty} y_n$ converges? It holds that $y_n \geq \frac{y_n}{1+y_n}$. If we would have to prove the converse we...

My problem : I have a function that I want to integrate, in the limit that a parameter goes to zero. I have a function ##f[x,r]## I want to compute ##F[r] = \int dx f[x,r]## and then series expand as ##r \rightarrow 0## This is impossible algebraically for me, but may be possible if I can...

Homework Statement The problem wants me to find the limit below using series expansion. ##\lim_{x \to 0}(\frac{1}{x^2}\cdot \frac{\cos x}{(\sin x)^2})## Homework EquationsThe Attempt at a Solution (1) For startes I'll group the two fractions inside the limit together ##\lim_{x...

Homework Statement Use zero- through third-order Taylor series expansion f(x) = 6x3 − 3x2 + 4x + 5 Using x0=1 and h =1. Once I found that the Taylor Series value is 49. I want to be able to check the value. On the board our teacher plugged in a value into the equation to show that the answer...

Hey guys, the question is 6.b. in the picture : http://imgur.com/FaKUMUZ Here is what I did to solve it : http://imgur.com/YrIvbTO I made these two simultaneous equations. 1875 comes from the fact that U1 + U2 = 1500 and U3 + U4 = 375. Then S4 must equal 1500+ 375(1875). I then found a formula...

I learn that we can expand the electric potential in an infinite series of rho and cos(n*phi) when solving the Laplace equation in polar coordinates. The problem I want to consider is the expansion for the potential due to a 2D line dipole (two infinitely-long line charge separated by a small...

In my book, applied analysis by john hunter it gives me a strange way of stating an infinite sum that I'm still trying to understand because in my calculus books it was never described this way. It says: We can use the definition of the convergence of a sequence to define the sum of an...

As the 2nd part of a question, we start with the Fourier sin series expansion of dirac delta function $\delta(x-a)$ in the half-interval (0,L), (0 < a < L): $ \delta(x-a) = \frac{2}{L} \sum_{n=1}^{\infty} sin \frac{n \pi a}{L} sin \frac{n \pi x}{L} $ The questions goes on "By integrating both...

Find the Fourier sin series expansion of dirac delta function $\delta(x-a)$ in the half-interval (0,L), (0 < a < L): Now $b_n = \frac{1}{L} \int_0^L f(x)sin \frac{n \pi x}{L}dx $ - but L should be $\frac{L}{2}$ for this exercise... So I would get $ \frac{2}{L} \int_0^L f(x)sin \frac{n \pi...

Homework Statement When two same lamps are connected with the same battery. Their lighting will be greater when they are connected in series or parallel? Homework Equations Series U=U1+U2+U3+... I=I1=I2=I3... Parallel U=U1=U2=U3... I=I1+I2+I3+... The Attempt at a Solution The answer is when...

Homework Statement Do the following series converge or diverge? ## \sum_{n=2}^\infty \frac{1}{\sqrt{n} +(-1)^nn}## and ##\sum_{n=2}^\infty \frac{1}{1+(-1)^n\sqrt{n}}##. Homework Equations Leibniz convergence criteria: If ##\{a_n\}_{k=1}^\infty## is positive, decreasing and ##a_n \to 0##, the...

Homework Statement Using the equality ##e = \sum_{k=0}^n \frac{1}{k!} + e^\theta \frac{1}{(n+1)!}## with ##0< \theta < 1##, show the inequality ##0 < n!e-a_n<\frac{e}{n+1}## where ##a_n## is a natural number. Use this to show that ##e## is irrational. (Hint: set ##e=p/q## and ##n=q##)...

Hello everyone, I need to calculate the radiant power of an interference pattern of a series of light wave reflections. I need a value in Watts that would plug in nicely into a photodetector's responsivity function (given in Amps/Watts) and thus giving me an estimation of the output current. I...

Mod note: Moved from Homework section I know that ##1/n^4## converges because of comparison test with ##1/n^2## (larger series) converges . how do I know ##1/n^2## converges? coz I cannot compare it with ##1/n## harmonic series as it diverges. @REVIANNA, if you post in the Homework & Coursework...

Hello everyone, I have a doubt about charging of capacitors in series. Suppose I connected two capacitors of same value, say,1 mF in series and put a bulb in series with them and applied a voltage 10V across the series. In steady state, the voltage across each capacitor will be 10/2=5V. Right...

Homework Statement The odd 2π-periodic function f(x) is defined by f(x) = x2 π > x > 0 -x2 −π<x<0 Find the coefficient bn in the Fourier series f(x) = a0/2 + ∑(an cos(nx) + bn sin(nx)). What are the values of the coefficients a0 and an and why? Homework Equations bn = 1/π ∫...

Mod note: Moved from Homework section 1. Homework Statement Understand most of the derivation of the E-L just fine, but am confused about the fact that we can somehow Taylor expand ##L## in this way: $$ L\bigg[ y+\alpha\eta(x),y'+\alpha \eta^{'}(x),x\bigg] = L \bigg[ y, y',x\bigg] +...

Power systems isn't my area of specialty and I've been doing some reading where it was stated that series voltage connections are safer than parallel connection. I don't fully understand why though. website address...

well, i have an calculus exam tomorrow and I'm 100% gona fail. I've neglected calculus so i could study for other subjects and left only 2 days to study taylor's polynomial aproximation, series and function series, the latter two are way more complicated than i expected. good thing is i can...

Homework Statement ∑n!/(3*4*5...*n) s1=1/3 sn=1/3+2/(4*3)+3!/(5*4*3)+...+n!/(3*4*5*...n) so i multiplied the sum with 1/2sn=1/6+1/(4*3)+1/(5*4)+1/(6*5)...+1/((n+2)(n-1)) got blocked here,i don't know how to continue, help please

Problem A now solved! Problem B: I am working with two equations: The first gives me the coefficients for the Laurent Series expansion of a complex function, which is: f(z) = \sum_{n=-\infty}^\infty a_n(z-z_0)^n This first equation for the coefficients is: a_n = \frac{1}{2πi} \oint...

Homework Statement Find the Laurent Series of f(z) = \frac{1}{z(z-2)^3} about the singularities z=0 and z=2 (separately). Verify z=0 is a pole of order 1, and z=2 is a pole of order 3. Find residue of f(z) at each pole. Homework Equations The solution starts by parentheses in the form (1 -...

Homework Statement An electric circuit consists of 3 identical resistors of resistance R connected to a cell of emf E and negligible internal resistance. What is the magnitude of the current in the cell? (in the diagram two of the resistors are in parallel with each other then the other in...

Hey, I am working on Calculus III and Analysis, I really need help with this one problem. I am not even sure where to begin with this problem. I have attached my assignment to this thread and the problem I need help with is A. Thank you!

Homework Statement A battery is connected with a resistor R1=4 om and then it is replaced with the resistance 9 om. In both cases the heat released in the same time is the same. Find the inner resistor of the battery. Homework Equations Q=UIt (U-tension; I-intensity, t-time) I=e.m.f/R+r The...

Homework Statement The interval of convergence of the Taylor series expansion of 1/x^2, knowing that the interval of convergence of the Taylor series of 1/x centered at 1 is (0,2) Homework Equations If I is the interval of convergence of the expansion of f(x) , and one substitutes a finite...

I learned about surface charge feedback theory in electrical circuits a few months ago and it has been extremely helpful for me to intuitively understand many concepts in electrical circuit analysis including conservative and non-conservative fields. I initially referred the paper written by...

Homework Statement A ball is rolling towards a rectangular hole which is 40cm deep and 2cm wide with a velocity 1m/s. It falls through the hole, bounces off the walls a couple of times and falls down. The direction of balls motion is perpendicular to the hole (falling in it from one side)...

Please help me find my mistake - "find the Sine F/series of f(x)=x over the half-interval (0,L)" I get $ b_n=\frac 2L \int_{0}^{L}x Sin \frac{2n\pi x}{L} \,dx $ $ = \frac 2L \left[ x(-Cos \frac{2n\pi x}{L}. \frac{L}{2n\pi x}\right] + \frac {1}{n\pi} \int_{0}^{L} Cos \frac{2n\pi x}{L} \,dx$...

Homework Statement : [/B] Given a voltage regulator with 6.8V Zener diode, input voltage range of 15-20V and load current range 5mA-20mA. Calculate the series resistance R for the regulator.Homework Equations : [/B] Applying KVL and no load situation, we get R = (V - Vz)/Iz where V is the...

Can anybody tell me how this is possible

Please help me with this Laurent series example for $\frac{1}{z(z+2)}$ in the region 1 < |z-1| < 3 Let w = z-1, then $ f(z) = \frac{1}{(w+1)(w+3)}=\frac{1}{2} \left[ \frac{1}{w+1}-\frac{1}{w+3} \right]$ I get $ \frac{1}{1-(-w)} = \sum_{n=0}^{\infty}(-1)^n w^n, \:for\: |w|<1;$ $ = -...

Hi - firstly should I be concerned that the dirac function is NOT periodic? Either way the problem says expand $\delta(x-t)$ as a Fourier series... I tried $\delta(x-t) = 1, x=t; \delta(x-t) =0, x \ne t , -\pi \le t \le \pi$ ... ('1' still delivers the value of a multiplied function at t)...

Hi - frustratingly I get some problems right 1st time, others just defy me (Headbang) $f(x) = -x, [-\pi,0]; = x, [0,\pi]$ I get $a_0 = \pi$ and $a_n = \frac{-4}{\pi \left(2n-1\right)^2}$ which agrees with the book - but I thought I'd check $b_n$ for practice, it should = 0 according to the...

Hi, appreciate some help with this FS problem - $f(t)= 0$ on $[-\pi, 0]$ and $f(t)=sin\omega t$ on $[0,\pi]$ I get $a_0=\frac{2}{\pi}$ and $b_1 = \frac{1}{2}$, which agree with the book; all other $b_n = 0$ because Sin(mx)Sin(nx) orthogonal for $m \ne n$ But $a_n...

Hi, in a section on FS, if I were given $\sum_{n=1}^{\infty} \frac{Sin nx}{n} $ I can recognize that as the Sin component of a Fourier Series, with $b_n = \frac{1}{n} = \frac{1}{\pi} \int_{0}^{2 \pi}f(x) Sin nx \,dx$ Can I find the original f(x) from this? Differentiating both sides doesn't...