Series Definition and 998 Threads

Evaluation of \displaystyle \sum_{r=1}^\infty \left(\frac{1}{36r^2-1}+\frac{2}{(36r^2-1)^2}\right)

I have recently been investigating summing divergent series and zeta function regularization's relation to dimensional re-normalization. Making some progress, but it is a bit slow despite literature being available...

Homework Statement The series is uniformly convergent on what interval? Homework EquationsThe Attempt at a Solution [/B] Using the quotient test (or radio test), ##|\frac{a_{n+1}}{a_{n}}| \rightarrow |x^2*\sin(\frac{\pi \cdot x}{2})|, n \rightarrow \infty##. However from here I'm stuck...

Homework Statement Show that ##\sum_{n=1}^{\infty}\frac{\log (1+1/n)}{n}## converges. Homework EquationsThe Attempt at a Solution If I take for granted the inequality ##\log (1+1/n) < 1/n##, I can easily show that this converges. My problem is is that I am not seeing how to prove convergence...

Hi, Before I post my question, let me admit that my foundation on mathematics is poor. I am trying to work on it, specifically on the application part. When I came through the following image, I was stuck to understand why I will need one like Taylor's series in a simple case like "F+ΔF = F...

Consider a straight duct silencer of any sort. If you place two of them in series, how will that affect the sound? If one silencer results in a 10 dB reduction for a specific frequency, will using two result in 20? It just seems like too much of an easy answer so I thought I would check with...

Homework Statement Determine for which ##r\not = 0## the series ##\displaystyle {\sum_{n=1}^\infty(2+\sin(\frac{n\pi}{3})) r^n}## converges. Homework EquationsThe Attempt at a Solution We have to split this up by cases based on ##r##. 1) Suppose that ##0<|r|<1##. Then...

Homework Statement Let ##r\in \mathbb{R}##. Determine whether ##\displaystyle \sum_{n=1}^{\infty}\sin \left(\frac{n\pi}{3}\right)\frac{1}{n^r}## converges Homework EquationsThe Attempt at a Solution If ##r>1##, then ##|\sin \left(\frac{n\pi}{3}\right)\frac{1}{n^r}| \le |\frac{1}{n^r}|##. The...

Homework Statement Prove that if ##(a_n)## is a decreasing sequence of positive numbers and ##\sum a_n## converges, then ##\lim na_n = 0## Homework EquationsThe Attempt at a Solution Let ##\epsilon >0##. By the Cauchy criterion there exists an ##N\in \mathbb{N}## such that ##\forall n\ge m\ge...

I know the result that if ##\lim a_{2n} = L = \lim a_{2n-1}##, then ##\lim a_n = L##. I'm wondering, can this be generalized to series? i.e., if ##\displaystyle \sum_{n=1}^{\infty}a_{2n-1}## and ##\displaystyle\sum_{n=1}^{\infty}a_{2n}## converge, then ##\displaystyle \sum_{n=1}^{\infty}a_{n}##...

Homework Statement Sorry for bad english[/B] There is serie of mass and pulleys. We Know we have more than two masses. System is balanced.We take m(m<<1) from on of the masses and add to another one. a) prove all the masses except these two have downward accleration b) prove mass with heavier...

Homework Statement Prove that for any finite group ##G## there exists a sequence of nested subgroups of ##G##, ##\{e\}=N_0\leq N_1\leq \cdots \leq N_n=G## such that for each integer ##i## with ##1\leq i\leq n## we have ##N_{i-1}\trianglelefteq N_i## and the quotient group ##N_i/N_{i-1}## is...

Homework Statement Suppose we are given the power series expansion ##f(x) = -\sum_{n=1}^{\infty} \frac{x^{n}}{n} ## which converges for |z|<1. What is the radius of convergence? Sum this serie and derive a power series expansion for the resulting function around -1/2, 1/2, 3/4 and 2. The...

When putting an LED in series with a Resistor do you put it according to current convention (positive to negative) or to the actual current flow? Does it matter?

Homework Statement Find the sum of ##\sum\limits_{n=2}^{\infty}\ln\left(1-\dfrac{1}{n^2}\right) ## Homework Equations No one. The Attempt at a Solution At first I though it as a telescopic serie: ##\sum\limits_{n=2}^{\infty}\ln\left(1-\dfrac{1}{n^2}\right) =\ln\left(\dfrac{3}{4}\right) +...

Homework Statement By finding a closed formula for the nth partial sum ##s_n##, show that the series ## s=\sum\limits_{n=1}^{\infty}nx^n## converges to ##\dfrac{x}{(1-x)^2}## when ##|x|<1## and diverges otherwise. Homework Equations Maybe ##s=\sum\limits_{n=0}^{\infty}x^n=\dfrac{1}{1-x}## when...

Above is a question I need some help with. I can get to a certain point, but I can't get beyond. Can somebody with knowledge about this help?

Hi All Been investigating lately ways to sum ordinarily divergent series. Looked into Cesaro and Abel summation, but since if a series is Abel Mable it is also Cesaro sumable, but no, conversely,haven't worried about Cesaro Summation. Noticed Abel summation is really a regularization...

I will state the problem below. I don't quite understand what I am needing to show. Could someone point me in the right direction? I would greatly appreciate it. Problem: Let p be a natural number greater than 1, and x a real number, 0<x<1. Show that there is a sequence $(a_n)$ of integers...

I've read that, in general, the energy of a wave, as opposed to what's commonly taught, isn't strictly related to the square of the amplitude. It can be seen to be related to a Taylor series, where E = ao + a1 A + a2A2 ... Also, that the energy doesn't depend on phase, so only even terms will...

Homework Statement Determine whether the series ## \frac {(n^3+3n)^{1/2}} {5n^3+3n^2+2 sin (n)}## converges or notHomework EquationsThe Attempt at a Solution looking at ## 1/sin (n) ## by comparison, ##1/n^2=1+1/4+1/9+1/16+...## converges for ##n≥1## for ##n≥1 ## implying that ##{sin (n)}≤n ##...

While I was was numerically integrating the magnetic field caused by an infinite array of magnetic moments, I observed the interesting limit ( limit (1) in the image). It may seem difficult to prove it mathematically but from the physics point of view, I think it can be proved relatively...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.14 ... ... Proposition 4.2.14 reads as follows...

Homework Statement One 18 watt lamp and two 60-watt light bulb are plugged into a 120V circuit. For either DC or AC, the two bulbs are connected each other in parallel and in series with the lamp in the same circuit. Calculate; i. the current flow through each light ii. the total...

Hello, I've been googling about this topic and have read from a number of different books but I still haven't found an exact answer to my question. It is known that the current is equal when the resistors are in a series. But the resistors per definition reduce current flow. If there are 2...

I am trying to solve an integral that has ##\frac{1}{1+x^2}## as a factor in the integrand. In my book it is claimed that if we use ##\displaystyle \frac{1}{1+x^2} = \sum_{n=0}^{\infty} (-1)^n \frac{1}{x^{2n+2}}## the problem can be solved immediately. But, I am confused as to where this series...

Homework Statement ##\displaystyle \int_0^1 \frac{\arctan x}{x}dx## Homework EquationsThe Attempt at a Solution I converted the integral to the following; ##\displaystyle \int_0^1 \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{2n+1}dx##. In this case am I allowed to swap the summation and integral signs?

Homework Statement Hello, I suspect this is an easy answer but I am not seeing it. I am reviewing (more so for fun / hobby) some differential equations – I’m not in school. I’m needing help with an example problem in Differential Equations With Boundary-Value Problems Zill 2nd edition. In...

I have the following series that I came up with in doing a problem: ##\displaystyle \sum_{n=0}^{\infty} \frac{1}{2^{n+1}(n+1)}##. I looked at WolframAlpha and it says that this series converges to ##\log (2)##. Is it possible to figure this out analytically?

Thank you for your time and effort. It is much appreciated. 1. Homework Statement I have attached the problem with the solution to this thread. Basically, the problem asks to construct the circuit model for a generic device by using the data of terminal current and voltage measurements. From...

Homework Statement The problem is attached as pic Homework Equations ∑(ƒ^(n)(a)(x-a)^n)n! (This is the taylor series formula about point x = 3)The Attempt at a Solution So I realized that we should be looking at either the 30th,31st term of the series to determine the coefficient. After we...

Homework Statement I am looking at the wikipedia proof of uniqueness of laurent series: https://en.wikipedia.org/wiki/Laurent_seriesHomework Equations look above or belowThe Attempt at a Solution I just don't know what the indentity used before the bottom line is, I've never seen it before...

Hi PF! One way to solve a simple eigenvalue problem like $$y''(x)+\lambda y(x) = 0,\\ y(0)=y(1)=0$$ (I realize the solution's amplitude can be however large, but my point here is not to focus on that) is to solve the inverse problem. If we say ##A[u(x)] \equiv d^2_x u(x)## and ##B[u(x)] \equiv...

Homework Statement I'm asked to find a combination of resistors (parallel and/or series) that uses resistors of 25 Ω, 100 Ω, 50 Ω, and 50 Ω. They should add up to give a total resistance of 62.5 Ω. Homework Equations Req for parallel = 1/R1 + 1/R2 + ... Req for series = R1 + R2 + ... The...

Homework Statement in title Homework EquationsThe Attempt at a Solution so i know that i have to use the ratio test but i just got completely stuck ((2x)n+1/(n+1)) / ((2x)n) / n ) ((2x)n+1 * n) / ((2x)n) * ( n+1) ) ((2x)n*(n)) / ((2x)1) * (n+1) ) now i take the limit at inf? i am stuck here i...

Homework Statement Homework EquationsThe Attempt at a Solution I don't understand why for the first part where the series goes up until arn-1, it cannot just go up until arn.. why will that first series always go up until arn-1 until it is multiplied by r?

Homework Statement What is the power series for the function ln (x+1)? How do you find the sum of an infinite power series? Homework Equations sigma from n=1 to infinity (-1)^n+1 (1/n2^n) That is the power series, how is that equivalent to ln (x+1)? How do you find the sum, or what does it...

Hi, I am a beginner and I don't speak very well... So I'm really sorry for my poor scientific language... I work on 1-Dimension time series of a same system measured at different periods. In these periods, time series have different chaotic characteristics as their lyapunov exponent are...

Homework Statement Three identical batteries are first connected in parallel to a resistor. The power dissipated by the resistor is measured to be P. After that, the batteries are connected to the same resistor in series and the dissipated power is measured to be 4P (four times larger than for...

Working on some microwave stuff, read about this but can't understand the explanations online.

Homework Statement S = 1+ x/1! +x2/2! +x3/3! +...+xn/n! To find S in simple terms. Homework Equations None The Attempt at a Solution I tried with Taylor's expansion, coshx and sinhx expansions. But cannot see consequence.

Homework Statement Homework EquationsThe Attempt at a Solution Not sure what I'm doing wrong here that looks like what the series is showing

Homework Statement Homework EquationsThe Attempt at a Solution I am not following what is going on here, how are they getting that part that is circled. i am just completely lost here

I'm looking at the proof of the alternating series test, and the basic idea is that the odd and even partial sums converge to the same number, and that this implies that the series converges as a whole. What I don't understand is why the even and odd partial sums converging to the same limit...

Homework Statement Homework EquationsThe Attempt at a Solution So the book is saying that this series diverges, i have learned my lesson and have stopped doubting the authors of this book but i don't understand how this series diverges. ok i can use the comparison test using 1/3n and 1/3n...

Homework Statement Homework EquationsThe Attempt at a Solution So my understanding of this so far is that the whole infinite series from 1 to infinities summation minus the first six terms summation is equal to 0.0002..? This is so confusing. So how does that mean that the sum will lie...

I don't understand something, the sum n=1 until infinity of (1/n) is a divergent harmonic series meaning that its sum is infinite right? After reading that i started thinking about the finite volume of the function (1/x) being revolved around the x-axis referred to as "Gabriels horn". They say...

Homework Statement ##f(x)=\sum_{n=0}^\infty x^n## ##g(x)=\sum_{n=253}^\infty x^n## The radius of convergence of both is 1. ## \lim_{N \rightarrow +\infty} \sum_{n=0}^N x^n - \sum_{n=253}^N x^n## 2. The attempt at a solution I got: ## \frac {x^{253}} {x-1}+\frac 1 {1-x}## for ##|x| \lt 1##...

Homework Statement [/B] Differentiate the power series for ##\frac 1 {1-x}## to find the power series for ##\frac 1 {(1-x)^2}## (Note the summation index starts at n = 1) 2. The attempt at a solution ##\sum_{n=1}^\infty n*x^{n-1}##

Homework Statement Homework Equations - The Attempt at a Solution Here's my work : However , the correct answer is : Can anyone tell me where's my mistake ?