Series Definition and 998 Threads

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)...

I have already got Faraday's "Experimental Researches in Electricity: Volume 1", which consists of 14 series of experiments concentrated in only one book. But, I wanted to see the more concentrated books, i.e the books of each series, to understand the situation then in elementary way. And even...

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;$ $ = -...

Show that this series diverges: $$\sum_{n = 0}^\infty \cos \left ( n^2 \right )$$ (in the sense that it takes arbitrarily large values as $n \to \infty$)

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...

Hi - an example in my book shows that FS coefficiants can be arrived at by minimizing the integrated square of the deviation, i.e. $ \Delta_p = \int_0^{2\pi}\left[ f(x) - \frac{a_0}{2}-\sum_{n=1}^{p}\left( a_nCosnx + b_nSinnx \right) \right]^2dx $ So we're looking for $ \pd{\Delta_p}{a_n}...

Need help. Determine the convergence of the series: 1. sum (Sigma E) from n=1 to infinity of: 1/((2*n+3)*(ln(n+9))^2)) 2. sum (Sigma E) from n=1 to infinity of: arccos(1/(n^2+3)) I think the d'alembert is unlikely to help here.

Hi folks, Just looking for an explanation on capacitor principles. My understanding: A capacitor is made from two conductors ( which have the ability to hold charge) separated by an insulator. Therefore current cannot flow between the+ and - plates. Unless unwanted breakdown from excessive...

Homework Statement How Can i do this on matlap the question in Attached files Homework Equations The Attempt at a Solution i try a lot but i failed

Hi there, I am reading through a thesis and the author takes the infinite series: \begin{equation} u(x,t)=u_0(x)+u_1(x)\cos(\sigma t - \phi_1(x)) + u_1'(x)\cos(\sigma' t - \phi'_1(x))+\ldots \end{equation} and by letting σr be the difference between the frequencies σ and σ' changes the above...

Homework Statement If we have a number sequence such that: a0, a1 are given, and every other element is given as ##a_n=\frac{(a_{n-1} + a_{n-2})}{2} then express an in terms of a0, a1 and n , and fin the limit of an Homework EquationsThe Attempt at a Solution If i try to express a3 in terms of...

Homework Statement Find the Fourier series defined in the interval (-π,π) and sketch its sum over several periods. i) f(x) = 0 (-π < x < 1/2π) f(x) = 1 (1/2π < x < π) 2. Homework Equations ao/2 + ∑(ancos(nx) + bnsin(nx)) a0= 1/π∫f(x)dx an = 1/π ∫f(x)cos(nx) dx bn = 1/π ∫f(x) sin(nx) The...

I was looking at the solution for problem 6 and I am confused on taking the derivatives of the function f(x)= cos^2 (x) I took the first derivative and did get the answer f^(1) (x)= 2(cos(x)) (-sin (x)), but how does that simplify to -sin (2x)? Is there some trig identity that I am not aware...

Hello,I've been reading my calculus book,and I can't tell the difference between a Taylor Series and a Taylor Polynomial.Is there really any difference? Thanks in advance

Hello,*please refer to the table above. I started from x(n)=x(n*Ts)=x(t)*delta(t-nTs), how can we have finite terms for discrete time F.S can anyone provide me a derivation or proof for Discrete F.S.?

If current is always the same in a series circuit then how is a transformer able to make the current smaller when it increases the voltage? is this just an exception since with the voltage being higher the same amount of power is being provided?

They ask for both $ \sum_{n=0}^{\infty} p^n Cos nx, also \: p^n Sin (nx) $ I'm thinking De Moivre so \sum_{n=0}^{\infty}p^n (e^{ix})^n = \sum_{n=0}^{\infty} p^n(Cos x + i Sin x)^n= \sum_{n=0}^{\infty} (pCos x + ip Sin x)^n I also tried a geometric series with a=1, $r=pe^{ix}$ But those...

Hi hi, So I worked on this problem and I know I probably made a mistake somewhere towards the end so I was hoping one of you would catch it for me. Thank you! Pasteboard — Uploaded Image Pasteboard — Uploaded Image

Hello, In finding a taylor series of a function using substitution, is it possible to use substitution for known taylor series of a function ,using different centers, and still get the same result. For example, if we have the function 1/(1+(x^2)/6) is it possible to use the taylor series of...

Homework Statement Periodic function P=3 f(t) = 0 if 0<t<1 1 if 1<t<2 0 if 2<t<3 a) Draw the graph of the function in the interval of [-3,6] b) Calculate the Fourier series of f(x) by calculating the coefficient. Homework EquationsThe Attempt at a Solution a) in attached...

Homework Statement Homework Equations The Attempt at a Solution Since P=2L, L=1 ? a_o = 1/2 [ ∫(from -1 to 0) -dx + ∫(from 0 to 1) dx ] = 1/2 [ (0-1) + (1-0) ] = 1/2(0) = 0 a_n = - ∫ (from -1 to 0) cosnπx dx + ∫ (from 0 to 1) cosnπx dx = 0 b_n = - ∫ (from -1 to 0) sinnπx dx...

Hi - my sometimes surprising set-book asks to show by series expansion, that $ \frac{1}{2}ln\frac{x+1}{x-1} =coth^{-1} (x) $ I get LHS = $ x+\frac{{x}^{3}}{3}+\frac{{x}^{5}}{5}+... $, which I think $= tanh^{-1} $ but I have found different expansions for the hyperbolic inverses, so I'd...

Hi, question asks to set upper and lower bounds on \sum_{n=1}^{1,000,000} \frac{1}{n} assuming (a) the Euler-Mascheroni constant is known and (b) not known. $\gamma = \lim_{{n}\to{\infty\left( \sum_{m=1}^{n} \frac{1}{m} \right)}} = 0.57721566$ and I found (a) easily (14.39272...), but no...

Hi, question is - show that the following series is convergent: $ \sum_{s}^{} \frac{(2s-1)!}{(2s)!(2s+1)}$ Hint: Stirlings asymptotic formula - which I find is : $n! = \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n $ I can see how this formula would simplify - but can't see how it relates to the...

Sorry for the bad English , do not speak the language very well. I posted this to know if the statement or " hypothesis " is correct . thank you very much =D. First Image:https://gyazo.com/7248311481c1273491db7d3608a5c48e Second Image:https://gyazo.com/d8fc52d0c99e0094a6a6fa7d0e5273b6 Third...

The text does it thusly: imgur link: http://i.imgur.com/Xj2z1Cr.jpg But, before I got to here, I attempted it in a different way and want to know if it is still valid. Check that f^{*}f is finite, by checking that it converges. f^{*}f = a_0^2 + a_1^2 cos^2x + b_1^2sin^2x + a_2^2cos^22x +...

Use the comparison test to see if \sum_{1}^{\infty}{\left[n\left(n+1\right)\right]}^{-\frac{1}{2}} converges? I tried n+1 \gt n, \therefore n(n+1) \gt n^2 , \therefore {\left[n(n+1)\right]}^{\frac{1}{2}} \gt n, \therefore {\left[n(n+1)\right]}^{-\frac{1}{2}} \lt \frac{1}{n} - no conclusion...

Homework Statement Homework Equations Series: R_{eq} = R_1 + ... + R_n Parallel: R_{eq} = \left(\frac{1}{R_1} + ... + \frac{1}{R_n} \right)^{-1} Charging Capacitor: I = I_0 e^{-t/RC} Charging Capacitor: \Delta V_C = \varepsilon (1- e^{-t/RC}) Charge: Q = C \Delta V_C The Attempt at a...

Homework Statement hello this question is discussed in 2009 but it is closed now If you invest £1000 on the first day of each year, and interest is paid at 5% on your balance at the end of each year, how much money do you have after 25 years? Homework Equations ## S_N=\sum_{n=0}^{N-1} Ar^n##...

Homework Statement In the arrangement shown,find the equivalent capacitance between A and B. Homework Equations Capacitance in parallel ##C##=##C_1##+##C_2##The Attempt at a Solution Supplied solution says As,we can clearly see that ,capacitors 10μF and 20μF are connected between same points...

Question: Why equations x(1-x)\frac{d^2y}{dx^2}+[\gamma-(\alpha+\beta+1)x]\frac{dy}{dx}-\alpha \beta y(x)=0 should be solved by choosing ##y(x)=\sum^{\infty}_{m=0}a_mx^{m+k}## and not ##y(x)=\sum^{\infty}_{m=0}a_mx^{m}##? How to know when we need to choose one of the forms. Also when I sum over...

Q. Show by induction that $ \sum_{1}^{\infty} \frac{1}{(2n-1)(2n+1)} = \frac{1}{2} $ So, start with base case n=1, $ S_1 = \frac{1}{(2-1)(2+1)} = \frac{1}{3}$? Maybe it's bedtime ...

Homework Statement All relevant data and variables are included in the image. The questions are also included in it. Homework Equations My questsion is just verification. I have attempted all the asked questions on the paper. Its frustrating as the papers don't include answers to check them...

1. Homework Statement Find the Fourier series of ##f(x) = \delta (x) - \delta (x - \frac{1}{2})## , ## - \frac{1}{4} < x < \frac{3}{4}## periodic outside. Homework Equations [/B] ##\int dx \delta (x) f(x) = f(0)## ##\int dx \delta (x - x_0) f(x) = f(x_0)##The Attempt at a Solution...

Homework Statement Approximate the integral to 3 decimal place accuracy via power series. ##\int_0^{1/2} x^2 e^{-x^2}\, dx ## Homework EquationsThe Attempt at a Solution ##x^2 e^{-x^2} = x^2 \sum_{n=0}^\infty \frac {(-x)^{2n}}{n!} = \sum_{n=0}^\infty \frac {x^{2n+2}}{n!}## ⇒ ##\int_0^{1/2}...

Homework Statement By considering the power series (good for |x| < 1) ##\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + x^4 +...## Describe how to manipulate this series in some way to obtain the result: ##\sum_{n=1}^\infty nx^n = \frac{x}{(1-x)^2}## Homework Equations Maclaurin...