Series Definition and 998 Threads

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...

Homework Statement ##\sum _{n=0}^{\infty }\:\sin \left(\frac{1}{n}\right)## Homework Equations The Attempt at a Solution Can I try comparison test by ##\left(\frac{1}{1+n}\right)<sin\left(\frac{1}{n}\right)## since ##\left(\frac{1}{1+n}\right)## diverges also...

Hey! :o I am looking at the following: Show that $\displaystyle{\text{exp}(1)=\sum_{k=0}^{\infty}\frac{1}{k!}=e}$ with $\displaystyle{e:=\lim_{n\rightarrow \infty}\left (1+\frac{1}{n}\right )^n}$. Hint: Use the binomial theorem and compare with the partial sum $s_n$ of the series...

Homework Statement hello, just came across this type of question for first time. A voltmeter with a range of 0-30volts is to be used to measure a 120 volt circuit. calculate the value of the resistor to be placed in series with the meter. the sensitivity of the meter is 1000 ohms per volt ...

Suppose we are asked to find the time period of vertical oscillations of this system. Then should we find the component of displacement along each spring and then add the forces by vector method or should we simplify the diagram into series and parallel connection like in electrical circuits and...

Hi, Is this possible to represent a periodic function like a triangular wave or square wave using a Taylor series? A triangular wave could be represented as f(x)=|x|=x 0<x<π or f(x)=|x|=-x -π<x<0. I don't see any way of doing although I know that trigonometric series could be used instead...

I'm kinda just hoping someone can look over my work and tell me if I'm solving the problem correctly. Since my final answer is very messy, I don't trust it. 1. Homework Statement We're asked to find the Fourier series for the following function: $$ f(\theta)=e^{−\alpha \lvert \theta \rvert}}...

Homework Statement Find the value of the sum (Infinity) Σ 2/((n+1)(n+3)) (n=1) Homework Equations Integral test Partial Sum Formula = k/2 (a_1 + a_k) The Attempt at a Solution Admittedly I started off this problem the wrong way. I used the integral test thinking I might get an answer...

The power series $$\sum_{n = 2}^\infty \frac{(n-1)(-1)^n}{n!}$$ converges to what number? So far, I've tried using the Ratio Test and the limit as n approaches infinity equals $0$. Also since $L<1$, the power series converges by the Ratio Test.

Homework Statement Prove the convergence of this series using the Comparison Test/Limiting Comparison Test with the geometric series or p-series. The series is: The sum of [(n+1)(3^n) / (2^(2n))] from n=1 to positive ∞ The question is also attached as a .png file 2. Homework Equations The...

While I was studying applications of Laplace transforms this thing showed up (lol). I seem to have a basic understanding of how the Zener model was derived. Seeing that the time-domain model apparently looks like a step function of sorts, I was trying to relate it to something like the behavior...

I understand algebraically that when capacitors are in series, the total capacitance is less than any individual capacitance, but I do not understand this intuitively. How can this be possible? Shouldn't more capacitors equal more capacitance?

Homework Statement In the following problem I am trying to extend the function $$f(x) = x $$ defined on the interval $$(0,\pi)$$ into the interval $$(-\pi,0)$$ as a even function. Then I need to find the Fourier series of this function.Homework EquationsThe Attempt at a Solution So I believe I...

Hello. I have this function ## v(x) = -\sum_{i=1} x^i \sqrt{2}^{i-2} \int_{-\infty}^{\infty} m^{i-1} \cosh(m)^{-4} dm## which I can not seem to figure out how to simplify.I tried looking at some partial integration but repeated integration of ## \cosh ## gives polylogarithms which seemed to...

In the following question I need to find the Fourier cosine series of the triangular wave formed by extending the function f(x) as a periodic function of period 2 $$f(x) = \begin{cases} 1+x,& -1\leq x \leq 0\\ 1-x, & 0\leq x \leq 1\\\end{cases}$$ I just have a few questions then I will be able...

Homework Statement With just about any problem asking for "rate at which source is delivering electrical energy to the circuit" or "find the power of the circuit" in a LRC circuit, I get that you have to calculate for the average power. But the multiple equations confuse me - sometimes in...

Homework Statement Find trigonometric Fourier series for ##f(x)=|x|##, ##x∈[−\pi, \pi]##, state points where ##F(x)## fail to converge to ##f(x)##. Homework Equations ##F(x) = \frac{a_0}{2}+\sum\limits_{n=1}^\infty a_ncos(\frac{n\pi x}{L})+b_nsin(\frac{n\pi x}{L})##...

Homework Statement Consider the Fourier series of a signal given by $$x(t)=\sum_{k=-\infty}^{\infty} a_ke^{jk\omega_0t}$$ Let's consider an approaches to this series given by the truncated series. $$x_N(t)=\sum_{k=-N}^{N} a_ke^{jk\omega_0t}$$ a- Show that if $x(t)$ is real then the series...

Let h(h(x)) = exp(x), where h(⋅) is holomorphic in the whole ℂ plane. I want an extension of the domain of exp(⋅) and of h(⋅) so that we can find values of these functions for x = Aleph(0).

Homework Statement I am having a slight issue with generating function of legendre polynomials and shifting the sum of the genertaing function. So here is an example: I need to derive the recurence relation ##lP_l(x)=(2l-1)xP_{l-1}(x)-(l-1)P_{l-2}## so I start with the following equation...

I'm trying to use Maxima to examine the error in a Fourier series as the number of terms increases. I've figured out how to produce a Fourier series and plot partial sums, but this has me stumped. If anyone experienced with the Maxima CAS has some insight into this, I would greatly appreciate...

Determine the sum: \[\frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}+\frac{12!}{16!}+...\]

Greg Bernhardt submitted a new PF Insights post A First Idea of Quantum Field Theory - 20 Part Series Continue reading the Original PF Insights Post.

Homework Statement Find Fourier coefficients of the periodic function whose template is x(t) where the Fourier Transform of x(t) is X(f) = (1-f^2)^2 where \left|f\right|<1 and period T_0= 4. Homework Equations FC=\hat x_T(k,T_0)=\sum_{k=-\infty}^\infty\frac{1}{T_0}X\left(k/T_0\right) The...

Homework Statement Hi! A battery of emf 12 V and negligible internal resistance is connected to a resistor of constant resistance 6 Ω, an ideal ammeter and an ideal voltmeter. The voltmeter and ammeter are in series with the cell and the resistor. What is the reading on each? Homework...

i have attached the problem set. I have done the first three problems but number 4 is very difficult. Can someone help me out? Thanks [Editor's note: The PDF below contains the complete problem set from which #4 is as shown above.]

I am trying to find a series representation for the following expression $$\int_{i=0}^\infty {x^{\frac{2n-1}{2}}(b+x)^{-n}}e^{\left(-{\frac{x^2}{2m}}+\frac{x}{p}\right)} dx$$ ; b,m,n,p are constant. Is there a name for this function? I found a series representation for $$\int_{i=0}^\infty...

Homework Statement i) What is the Taylor Series for f(x) = (1+x)^m about x=0 where m is a real number? ii) Why does this binomial series terminate when m is a non-negative integer? A iii) Can the result to (i) be used to find the first four non-zero terms of the series for (1+x)^(-1/2)...

Homework Statement Find ∫qk(x) dx where the upper bound is 1 and the lower bound is 0. g is some function and we are finding for k = 2,6,10 and 14, hence the first four non-zero terms of a series that can be used to calculate approximations to I = ∫sin(x^2) dx were the upper bound is 1 and the...

https://drive.google.com/file/d/0B0NXDy0RMDe7MXhMcjZBdkhoSDg/view?usp=sharing PIC: [https://drive.google.com/file/d/0B0NXDy0RMDe7MXhMcjZBdkhoSDg/view?usp=sharing] 1. A rotating disk is connected with two arms AD and DB which are rotating with the rate of 0.2 rad/s^2 and -0.3 rad/s^2...

Hello Everyone ! I am interesting to find descriptions of the series of experiments that Newton made for determining the laws of motion. In English of course.

Homework Statement Find the Taylor series for: ln[(x - h2) / (x + h2)] Homework Equations f(x+h) =∑nk=0 f(k)(x) * hk / k! + En + 1 where En + 1 = f(n + 1)(ξ) * hn + 1 / (n + 1)! The Attempt at a Solution ln[(x - h2) / (x + h2)] = ln(x-h2) - ln(x + h2) This is as far as I have been able to...

I'm trying to make an approximation to a series I'm generating; the series is constructed as follows: Term 1: \left[\frac{cos(x/2)}{cos(y/2)}\right] Term 2: \left[\frac{cos(x/2)}{cos(y/2)}-\frac{sin(x/2)}{sin(y/2)}\right] I'm not sure yet if the series repeats itself or forms a pattern...

Homework Statement One type of tuning circuit used in radio receivers is a series LCR circuit. You like listening to a station1 that transmits 99.3 MHz in . The government wants to make 100.1 MHz available to station2. Assume that the transmitters of the two stations are equally powerful and...

Svein submitted a new PF Insights post Further Sums Found Through Fourier Series Continue reading the Original PF Insights Post.

Hi, First of all, I want to say that I know how can define and calculate Fourier coefficients but I have some question about the final presentation of Fourier and half-period or unknown period functions. 1)In this function how can we define T? 2)for above diagram, in a book, they define f(t)...

Homework Statement f(x)=x on [0,2) Homework Equations Fourier Series is given as: f(x)=a0/2 + n=1∞∑(an*cos(nπx/L) + bn*sin(nπx/L) a0=1/L*-LL∫f(x)dxThe Attempt at a Solution Basically what I am being taught is that we take the Period, T, to be equal to 2L so, T=2L In this case T=2 and L=1. My...

Hello I'm working through a book (with answers) but am struggling with voltage, current, resistance and circuits. Please check my understanding below and let me know if I've finally understood. Thank you. In particular I'm confused in Q2 Q1. A student connects light bulbs, A and B, and...

Homework Statement Consider a circuit that consists on a resistor of an intrinsic semiconductor R and a capacitor C in series. The voltage between the terminals of the circuit is U, which is an alternated sinusoidal voltage. U1, which is the voltage in the capacitor as a phase difference of 30...

Is it ok to assume that the entropy ##S## of an arbritary system can be written as a power series as a function of the system's internal energy ##U##? Like $$S(U) = \sum_{i=1}^{\infty}a_iU^i = a_1 U + a_2 U^2 + \ ...$$ with ##a_i \in \mathbb{R}##. What results could be obtained from such...

Homework Statement Given ##b_n = 1 / n## if ##n## odd and ##b_n = 1 / n^2## if ##n## even, show that the series $$\sum_{n=1}^{\infty} (-1)^n b_n$$ diverges. Homework Equations Did'nt find any for this problem The Attempt at a Solution I assumed that ##\sum_{n=1}^{\infty} (-1)^n b_n =...

Homework Statement (see my attached photo to better understand where I am coming from!) So after some research, I've discovered that the current at different points in a simple series circuit is supposed to be the same value, and that the voltage is supposed to be different values. I...

Hi guys! (see my attached photo to better understand where I am coming from!) So after some research, I've discovered that the current at different points in a simple series circuit is supposed to be the same value, and that the voltage is supposed to be different values. I performed a lab on...

Homework Statement Assume that ##a_k > 0## and ##\sum_{k=0}^\infty a_n## converges. Then for every ##\epsilon > 0##, there exists a ##n \in Bbb{N}## such that ##\sum_{k=n+1}^\infty a_k < \epsilon##. Homework EquationsThe Attempt at a Solution Since the series converges, the sequence of...

Suppose we have monthly totals of observed data for last 35 years. That data is of inflow of a river in a reservoir and monthly demands from the reservoir. We are interested to check the effect of construction of a dam in the upstream. The effect is, whether the downstream reservoir will have...

Homework Statement In Complex Fourier series, how to determine the function is odd or even or neither, as in the given equation $$ I(t)= \pi + \sum_{n=-\infty}^\infty \frac j n e^{jnt} $$Homework Equations ##Co=\pi## ##\frac {ao} 2 = \pi## ##Cn=\frac j n## ##C_{-n}= \frac {-j} n ## ##an=0##...

Homework Statement Homework Equations The Attempt at a Solution a0=4 an=8/Pi*n Heres a written solution https://gyazo.com/57e11d1e7a360914db8aec167beb6b39.png