Series Definition and 998 Threads

Homework Statement Use the power series method to solve the initial value problem: ##(x^2 +1)y'' - 6xy' + 12y = 0, y(0) = 1, y'(0) = 1## Homework EquationsThe Attempt at a Solution The trouble here is that after the process above I end up with ##c_{k+2} = -...

When using the method of differences on a given series, when do you stop listing the terms? Example question: f(r)= ; r∈N State f(r)-f(r+1) in terms of r and hence determine So skipping until the worked answer gives Great so here I included the n+1th term because I'm guessing since the...

Given that the sum of the first n terms of series, s, is 9-32-n Find the sum of infinity of s. Do I use the formula S\infty = \frac{a}{1-r}?

Given that the sum of the first n terms of series, s, is 9-32-n show that the s is a geometric progression. Do I use the formula an = ar n-1? And if so, how do I apply it?

Hey! :o I want to show that series $$f(x)=\sum_{k=1}^{\infty}2^k\sin (3^{-k}x)$$ is continuously differentiable. We have that $|2^k\sin (3^{-k}x)|\leq 2^k\cdot 3^{-k}=\left (\frac{2}{3}\right )^k$, or not? The sum $\sum_{k=1}^{\infty}\left (\frac{2}{3}\right )^k$ converges as a geometric...

Given that the sum of the first n terms of series, s, is 9-3^2-n (i) find the nth term of s. Do I have to use the formula sn = a(1-r)/1-r?

Homework Statement Show that $$\frac{(-1)^nn!}{z^n}$$ is divergent. Homework Equations We can use the ratio test, which states that if, $$\lim_{n\to\infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|>1$$ a series is divergent. The Attempt at a Solution Applying the ratio test, we find that...

Homework Statement Homework Equations All I know is the a's have something to do with the integrals. The Attempt at a Solution I used FFT analysis in Matlab but I do not know what I am looking for. How do the a0s relate to the f(t) in the question and how would I even do run that equation in...

Homework Statement http://sites.math.rutgers.edu/~ds965/temp.pdf (NUMBER 2)[/B]Homework Equations I do not understand the alternating part for the second problem and the recursive part for the first problem.The Attempt at a Solution The first answer I got was first by writing out the...

we know the five spectral series of Lyman, Balmer, Paschen, bracket, and Pfund their wavelength and also the part of EM spectrum they fall in, my question is why do we neglect the 6th series in the spectrum? and in what part of EM spectrum the 6th series exist and what could be its wavelength...

Homework Statement Homework Equations Power series ODE The Attempt at a Solution [/B] Sorry for not typing all those things out from my phone.. How can I get C1? And how can I put the solution in the required format? (I don't know how to put it in summation sign... and i cannot even solve...

Homework Statement Homework Equations Power series The Attempt at a Solution As I have to write in form of "x^2n" & "x^2n+1", I am totally have no idea with how can I go on to do the question. Those I have learned in lecture and online are mostly with only one part of summation... or two...

Homework Statement ##\sum_{n=1}^{\infty }1+(-1)^{n+1} i^{2n}## Is this series divergent or convergent? Homework Equations 3. The Attempt at a Solution [/B] I tried using the divergent test by taking the limit as ##n## approaches ##{\infty }##, but both ##i^{2n}## and ##(-1)^{n+1}## will...

If I had a simple series circuit with only a single resistor, and I used a voltmeter to find the voltage between a point at the end of the circuit and another point, which was moved from the beginning to the end of the circuit, what would I find at these various point? Would the voltage remain...

Homework Statement https://imgur.com/DUdOYjE The problem (#58) and its solution are posted above. Homework Equations I understand that I can approach this two different ways. The first way being the way shown in the solution, and the second way, which my professor suggested, being a Direct...

The problem In this problem I am supposed to show that the following series converges by somehow comparing it to ## \frac{1}{k\sqrt{k}} ## : $$ \sum^{\infty}_{k=1} \left( \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} \right) $$ The attempt ## \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} =...

If one hooks a 1154 and an old 6 volt automotive headlamp in series and powers it with 12 volts will the headlamp being that it is higher wattage cause all the current flow thru the smaller wattage bulb causing it to burn out?

Homework Statement Determine whether the following series converge, converge conditionally, or converge absolutely. Homework Equations a) Σ(-1)^k×k^3×(5+k)^-2k (where k goes from 1 to infinity) b) ∑sin(2π + kπ)/√k × ln(k) (where k goes from 2 to infinity) c) ∑k×sin(1+k^3)/(k + ln(k))...

Hello, I tried to compute the Fourier series coefficients for the Dirac comb function. I did it using both the "complex" formula and the "real" formula for the Fourier series, and I got : - complex formula : Cn = 1/T - real formula : a0 = 1/T, an = 2/T, bn = 0 This seems to be valid since it...

Can anyone tell me how if the derivative of n(n') is quadratic the second term in the taylor series expansion given below vanishes. This doubt is from the book Classical Mechanics by Goldstein Chapter 6 page 240 3rd edition. I have attached a screenshot below

I need help understanding what will happen when the switch in closed in this circuit. What I want to happen is for Cap B to charge first and then discharge into Cap C. When the charged capacitor begins to discharge, will it charge Caps B and C at the same time? It will have to overcome the...

I'm designing a device that consumes 450nA in idle (up to 2mA peak) and the maximum allowed voltage is 3.3V. The power is supplied by a battery with a voltage of 3.6V. One of the problems I've run into is: All LDOs I could find have a quiescent current consumption (Iq) greater than 450nA. The...

Homework Statement Homework EquationsThe Attempt at a Solution I considered N=2 . Two similar charged capacitors are joined in series i.e positive plate of one is joined to negative of the other . If I consider that there is no movement of charge since both the capacitors are similar and...

Post moved by moderator, so missing the homework template. series ##{a_n}## is define by ##a_1=1 ## , ##a_2=5 ## , ##a_3=1 ##, ##a_{n+3}=a_{n+2}+4a_{n+1}-4a_n ## ( ##n \geq 1 ##). $$\begin{pmatrix}a_{n+3} \\ a_{n+2} \\ a_{n+1} \\ \end{pmatrix}=B\begin{pmatrix}a_{n+2} \\ a_{n+1} \\ a_{n} \\...

I have a time series model constructed by using ordinary least square (linear). I am supposed to provide some general comments on how one would improve the robustness of the analysis of a time series model (in general). Are there any general advice apart from expanding data, making it more...

Homework Statement Test the series for convergence or divergence ##1/2^2-1/3^2+1/2^3-1/3^3+1/2^4-1/3^4+...## Homework Equations rn=abs(an+1/an) The Attempt at a Solution With some effort I was able to figure out the 'n' th tern of the series an = \begin{cases} 2^{-(0.5n+1.5)} & \text{if } n...

I am having trouble describing the function that I am taking sum of in a series. Like in the example \begin{equation} \sum_{z=0}^{\infty}f(z)\end{equation}: What would I call $f(z)$? Would it be the argument of the series?

Homework Statement Find the Fourier series of the function ##f## given by ##f(x) = 1##, ##|x| \geq \frac{\pi}{2}## and ##f(x) = 0##, ##|x| \leq \frac{\pi}{2}## over the interval ##[-\pi, \pi]##. Homework Equations From my lecture notes, the Fourier series is ##f(t) = \frac{a_0}{2}*1 +...

I already learn to use Taylor series as: f(x) = ∑ fn(x0) / n! (x-x0)n But i don´t see why the serie change when we use differents x0 points. Por example: f(x) = x2 to express Taylor series in x0 = 0 f(x) = f(0) + f(0) (x-0) + ... = 0 due to f(0) = (0)2 to x0=1 the series are...

Arithmetic Series? Given the arithmetic series 5+14+23+...(to 241 terms), how many terms in the series are divisible by 5? I need a good explanation and a good start.

This is related to steel wire pulling machine. I have 2 cases. In CASE-1: ONE MOTOR 160 kW AND CASE-2: TWO MOTORS 129 kW Each. i.e 2x129 kW Which case would have an efficient pulling force and efficiency Case 1 or Case 2.

Homework Statement Determine if the series is convergent. Homework Equations ∞ ∑ (((2n^2 + 1)^2)*4^n)/(2(n!)) n=1[/B] The Attempt at a Solution I'n using the Ratio Test and have got as far as (4*(2(n+1)^2+1)^2)/((n+1)((2n^2+1)^2)). I know this series converges but I need to find the...

Homework Statement I am only interested in 9 (a) Determine the Fourier Cosine series of the function g(x) = x(L-x) for 0 < x < L Homework Equations The Answer for 9 a. g(x) = (L^2)/6 - ∑(L^2/(nπ)^2)cos(2nπx/L) This is the relevant equation given where ω=π/L f(t) = a0+∑ancos(nωt) a0=1/L...

Homework Statement a. What is the net resistance in the circuit? b. What is the current through the 4 Ω resistor? c. What is the voltage drop across the 3 Ω resistor? Homework Equations R12 = R1 +R2 R34 =R3 + R4 1/Rt = 1/R12 + 1/R34 I3=I4 The Attempt at a Solution a) R12 = 2+ 3...

Homework Statement Show that the sequence of partial sums s_{n} = 1+\sum_{i=1}^{n} \left(\prod_{k=1}^{i}\left( \frac{1}{2} + \frac{1}{k}\right)\right) converges, with n\in \mathbb{N}\cup \{0\} Homework EquationsThe Attempt at a Solution [/B] So we want to find \lim_{n\to\infty} s_{n} =...

MinutePhysics is attempting to produce a series of video lessons on Special Relativity, using an approach, according to the video, that will be different and "simpler" than the traditional method that SR has been taught in schools. Since we often get questions on here about this topic...

Homework Statement There are 2 circuits. A: -A series circuit Components: -Motor -Filament lamp -Resistor B: -A parallel circuit Components: -Motor -Filament lamp -Resistor -Each component is in a separate parallel circuit Question)Explain why the power of the motor is lower in the circuit...

Homework Statement Consider a power series solution about x0 = 0 for the differential equation y'' + xy' + 2y = 0. a) Find the recurrence relations satisfied by the coefficients an of the power series solution. b) Find the terms a2, a3, a4, a5, a6, a7, a8 of this power series in terms of the...

Find the exact sum of the series: $$S = \frac{1}{1\cdot 2\cdot 3\cdot 4}+\frac{1}{5 \cdot 6 \cdot 7 \cdot 8}+...$$

Im reading that very hot stars and very cool stars have weak hydrogen lines. With that being said, how do we know that these very hot stars contain high quantity of hydrogen if we can't see it in the spectra?

Homework Statement Find Laurent series of $$z^2sin(\frac{1}{1-z})$$ at $$0<\lvert z-1 \rvert<\infty$$ Homework Equations sine series expansion. The Attempt at a Solution At first, it seems pretty elementary since you can set w=\frac{1}{z-1} and expand at infinity in z, which is 0 in w...

Does the complex form of Fourier series assume even or odd half range expansion?

Homework Statement Hello, I have a general question regarding to coefficient matching when spanning some function, say , f(x) as a linear combination of some other basis functions belonging to real Hilbert space. Homework Equations - Knowledge of power series, polynomials, Legenedre...

Hi, I'm aware that the total resistance in a series connection is the sum of all the resistors involved, and that the current is the same throughout, and that the voltage will be different for each resistor but the total voltage will be their sum as well. However, I would like to inquire, does...

Hi, I keep reading in multiple sources that amplifier output can be given by Vout = a0 + a1v(t) + a2v2(t) + a3v3(t) + ... + anvn(t) I've checked in three of my textbooks and there is not a clear definition (its often just stated) why this equation is used and why it works. I am not looking...

Hey! :o I want to show that $\displaystyle{\lim_{x\rightarrow \infty}\frac{e^x}{x^{\alpha}}=\infty}$ and $\displaystyle{\lim_{x\rightarrow \infty}x^{\alpha}e^{-x}=0}$ using the exponential series (for a fixed $\alpha\in \mathbb{R}$). I have done the following: $$\lim_{x\rightarrow...

I am asking on the spur, so there has not been too much thought put into it, but how would we classify a series summation such as $$ \sum_{i=0}^{n} 2^{2^i} ~ ?$$ It does not feel to be geometric, nor that it can be made to be geometric. In general, the function xx does not look like it bears a...

We have: Period T = 4, so fundamental frequency w0 = pi/2. This question seems sooo easy. But when I use the integral: x(t) = Σa[k] * exp(i*k*pi/2*t). I get 1 + sum(cos(k*pi/2*t)), which does not converge. Where did I went wrong ? Thanks a lot for your help.

hey I am trying to calculate the limit of : limn→∞(1/2+3/4+5/8+...+2n−1/2^n) but I am not sure how to solve it, I thought to calculate 2S and than subtract S, but it did not worked well. I did noticed that the denominator is a geometric serie,but I don't know how to continue. could you help?

Homework Statement There is a sawtooth function with u(t)=t-π. Find the Fourier Series expansion in the form of a0 + ∑αkcos(kt) + βksin(kt) Homework Equations a0 = ... αk = ... βk = ... The Attempt at a Solution After solving for a0, ak, and bk, I found that a0=0, ak=0, and bk=-2/k...