Series Definition and 998 Threads

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...

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...

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...

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

I watched Hellraiser: Bloodlines last night about the box that can open the gateway to hell. This is very interesting series and lines of movies. Anything similar to it you recalled? Scientists explore concepts of other multiverses, other brane dimensions now and it would be thrilling if...

It has been defined that for an alternating series, the difference between the total sum of the series and the partial sum of the series through nth term is always less than or equal to the (n+1)th term. Can anyone explain the intuitive reason behind this?

I am on book IV Wizard and Glass. Anyone else read these? I am interested in seeing The Dark Tower movie. It's one of the few science fiction series that has got my interest.

Homework Statement Q:/ Find the complex form of Fourier series for the following periodic function whose definition in one period is given below then convert to real trigonometry also find f(0). f(t)=cos(t/2), notes: (T=2*pi) (L=pi) Homework Equations 1) f(t)=sum from -inf to +inf (Cn...

Hi guys, how do I figure out whether components are in series/parallel with each other just by looking at them? for example, in the circuit on the picture I posted, how do I know which of the resistors are in series/ parallel with each other? I know the definition involves nodes, but in clear...

Homework Statement Show that the series ##\displaystyle \sum_0^{\infty} (-1)^nn## diverges Homework EquationsThe Attempt at a Solution It seems obvious that it diverges, for although the terms oscillate, they get bigger and bigger and never really cancel each other out. However, I am not sure...

I have a hard time comprehending excess charge in capacitor plates. Let's say we have two identical capacitor and we charged them to identical amount of charge. Next we connect them in series by opposite sign electrodes and we get double the voltage on the open ends. Now a single capacitor has...