Series Definition and 998 Threads

##\sum_{n=1}^\infty 1/n^2 ## converges to ##π^2/6## and every other series with n to a power greater than 1 for n∈ℕ convergesis it known if the sum of all these series - ##\sum_{m=2}^\infty \sum_{n=1}^\infty 1/n^m ## for n∈ℕ converges? apologies for any notational flaws

Homework Statement Determine whether the given series is absolutely convergent, conditionally convergent, or divergent. ##\frac{1}{3} + \frac{1 \cdot 4}{3 \cdot 5} + \frac{1 \cdot 4 \cdot 7}{3 \cdot 5 \cdot 7} + \frac{1 \cdot 4 \cdot 7 \cdot 10}{3 \cdot 5 \cdot 7 \cdot 9} + \ldots + \frac{1...

Homework Statement Hello. I'm not entirely sure what this question is asking me, so I'll post it and let you know my thoughts, and any input is greatly appreciated. If the series ##\sum_{n=0}^\infty a_n(x-4)^n## converges at x=6, determine if each of the intervals shown below is a possible...

Reading this piece with a number of proofs of the divergence of the harmonic series http://scipp.ucsc.edu/~haber/archives/physics116A10/harmapa.pdf and this example states: 'While not completely rigorous, this proof is thought-provoking nonetheless. It may provide a good exercise for students...

Greg Bernhardt submitted a new blog post The Sum of Geometric Series from Probability Theory Continue reading the Original Blog Post.

One can have a progression and it is called a Sequence. One can sum the terms in a sequence or progression, and this is called a Series. Why those terms like that; or why those two different terminologies? Was it decided just to pick a word Series so as to avoid the need to use Sum Of the...

I've seen the proof that the sum of 1/n for = 1 to infinity is infinity (which still blows my mind a little). Is the sum of 1/nn for n = 1 to infinity also infinity? i.e, 1 + 2/4 + 3/27 + 4/256+...

Homework Statement Use the integral test to compare the series to an appropriate improper integral, then use a comparison test to show the integral converges or diverges and conclude whether the initial series converges or diverges. ##\sum_{n=3}^\infty \frac{n^2+3}{n^{5/2}+n^2+n+1}## Homework...

Homework Statement Obtain Maclaurin Series for: f(x) = sin(x2)/x Homework Equations f(x) = ∑f(n)(c) (x-c)n / n! (for Maclaurin c = 0) The Attempt at a Solution I know that sin(x2) = x2 - (x2*3/3! +... from the final answer I see, that this is just multiplied to 1/x. This bothers me...

Homework Statement - Given a bounded sequence ##(y_n)_n## in ##\mathbb{C}##. Show that for every sequence ##(x_n)_n## in ##\mathbb{C}## for which the series ##\sum_n x_n## converges absolutely, that also the series ##\sum_n \left(x_ny_n\right)## converges absolutely. - Suppose ##(y_n)_n## is...

I know this may sound as a stupid question but I would like to clarify this. An arbitrary function f can be expressed in the Fourier base of sines and cosines. My question is, Can this method be used to solve any differential equation? You plug into the unkown function the infinite series and...

Hi, I've computed 512 terms of a power series numerically. Below are the first 20 terms. $$ \begin{align*} w(z)&=0.182456 -0.00505418 z+0.323581 z^2-0.708205 z^3-0.861668 z^4+0.83326 z^5+0.994182 z^6 \\ &-1.18398 z^7-0.849919 z^8+2.58123 z^9-0.487307 z^{10}-7.57713 z^{11}+3.91376 z^{12}\\...

Hi Everyone, I am currently working on a project where I am creating English subtitles for an Italian TV Series, for deaf and hard of hearing. There is one particular line I am struggling to hear and it is where one of the characters is talking about a type of physics. I wondered if anyone...

Homework Statement The battery is disconnected from a series RC circuit after the capacitor is fully charged and is replaced by an open switch. When the switch is closed, a. the capacitor does not allow current to pass b. the current stops in the resistor c. the potential difference across the...

I just finished watching this biographical drama on Einstein's life. I found it quite good. Has anyone else here seen it?

ETA. Read the bottom post first. Well, and.. Obviously mathematicians know this identity. At the x=b=c=n=2 point, pi exists. There are also connections to the Wallis product (pi/2). Anyway, I simplified it to the n=2 case. And re-remembered my fascination with the Pidentity, where...

I have attached a picture of what I want Matlab to do. I basically want Matlab to show the list of independent variable 'n' and then another column showing the terms when n=0...10. Some of the outputs are in variable form and others in numerical form. My attempt so far is stated below. I have no...

My attempt I used negative binomial to solve the problem, however I'm left with a polynomial that is difficult to solve? Is there any other way to approach this problem? I used the inequality because I'm trying to find the range of p. Since the probability of winning the series for team...

1) Are there any periodic alternating series functions other than sine and cosine (and series derived from them, like the series for cos(a) * cos(b))? 2) What is the following series called when x is (0,1) and (1,2]? Quasiperiodic? Semi? \sum_{n=0}^\infty \, (-1)^n \...

Homework Statement Show that the magnitude of the net force exerted on one dipole by the other dipole is given approximately by:$$F_{net}≈\frac {6q^2s^2k} {r^4}$$ for ##r\gg s##, where r is the distance from one dipole to the other dipole, s is the distance across one dipole. (Both dipoles are...

We were informally introduced Taylor series in my physics class as a method to give an equation of the electric field at a point far away from a dipole (both dipole and point are aligned on an axis). Basically for the electric field: $$\vec E_{axis}=\frac q {4πε_o}[\frac {1} {(x-\frac s 2)^2}-...

Hello there, I am studying Taylor series, and in the slides given to us we calculated the taylor series of ln $(\frac{1+x}{1-x} )$ = ln(1 + x) − ln(1 − x), by using standard Taylor series of ln(1 + x). The notes then proceed to say : " It can be shown that every positive real number t can be...

Does a series circuit have branches? or the term branch can use in series circuit too and there are only one branch in series circuit. please explain.

Hi, This is not homework. I am reading a book: "The art of infinite: The Pleasure of Mathematics" and pages 66-67 discuss a series and the author seems to be making some assumptions that are not clear to me. So I want to make sure that I understand this. 1. Homework Statement A 3-rhythm...

Homework Statement The equation of motions of a series of pendulums coupled by a torsion spring is this: ##\ddot{\Phi_i}=-\frac{k}{ml^2}(2\Phi_i-\Phi_{i-1}-\Phi_{i+1})##, where k is the torsion spring constant, m is the mass of a single pendulum, and l is the length of a single pendulum. We...

Homework Statement ##\sum_{k=0}^\infty \frac 4 k(\ln k)^2 ## Homework EquationsThe Attempt at a Solution I tried to solve it using the integral test but since it's not continuous it doesn't work.

Homework Statement Homework EquationsThe Attempt at a Solution My attempted solution is above and here https://imgur.com/8RmDMf8/ I'm confused as to the answers in the book being i and iii (I just don't see how i is included). If critical damping occurs at the value above, and if you go above...

If I have a limit of a series then how can I convert it into integral. I know to convert a sum into an integral there must be Δx multiplied to each term and this must go zero. Can you please explain me the conversion of limit of series (normal series with no Δx) into an integral. Thank you.

Hello! (Wave) Let $0< \theta<1$ and a sequence $(a_n)$ for which it holds that $$|a_{n+2}-a_{n+1}| \leq \theta |a_{n+1}-a_{n}|, n=1,2, \dots$$ We have already shown that $(a_n)$ converges. Could you give me a hint how we could also show that $\sum_{n=1}^{\infty} (a_{n+1}-a_n)$ converges?

Evaluate the indefinite integral as a power series ∫[ln(1−t)/7t]dt. Find the first five non-zero terms of power series representation centered at t=0. Answer: f(t)= What is the radius of convergence? Answer: R= 1 Note: Remember to include a constant "C". This problem has been difficult...

Hi. What books would be good to complement the Greiner theoretical physics series? Greiner covers Newtonian mechanics, analytical mechanics, electrodynamics, thermodynamics, statistical mechanics, quantum mechanics (at great length), relativistic quantum mechanics, quantum field theory...

Homework Statement Hello, I need some feedback on whether this reasons is correct. consider the series Examine the series for absolute convergence. Homework EquationsThe Attempt at a Solution How I have solved this, using the limit comparison test: we have: introducing we have that...

Homework Statement I need to solve the DE y’ = x^2y using the power series method Homework Equations y = sum(0->inf)Cnx^n y’ = sum(1->inf)nCnx^(n-1) The Attempt at a Solution I plug in the previous two equations into the DE. What is the general procedure for these problems after that...

Homework Statement Find the sum of the series Homework EquationsThe Attempt at a Solution Not sure exactly where to start. If I move 3 outside the sum I'm left with 3*sigma(1/n*4^n), which I can rewrite to 3*sigma((1/n)*(1/4)^n), which party looks like a geometric series..Any tips?

Hi, I have a graduate level understanding of power electronics. The other day I decided I wanted to series the output of two DC Boost converters. I remembered the simple conceptual circuit of how they worked, and I jumped onto google to see if this could be done safely. What I read was that it...

I am reading Stephen Abbott's book: "Understanding Analysis" (Second Edition) ... I am focused on Chapter 6: Sequences and Series of Functions ... and in particular on power series ... I need some help to understand Theorem 6.5.1 ... specifically, some remarks that Abbott makes after the proof...

Homework Statement Hello, I need to find an expression for the sum of the given power series The Attempt at a Solution I think that one has to use a known Maclaurin series, for example the series of e^x. I know that I can rewrite , which makes the expression even more similar to the...

This is a rather simple question, but am I understanding the following correctly? 1. Homework Statement The Attempt at a Solution This isn't really the problem, but I have a feeling my problem the assignments, is me misunderstanding the function description. I don't see how this 2 pi...

Homework Statement The problem is given in the attached picture, but I already have a solution to part a) which I am confident in (I have checked it carefully, compared to other students and confirmed it with my graduate-TA). Part b) asks us to plot the equipotentials but I cannot figure out...

Homework Statement So I'm really confused with mutual inductors and dot convention. If your answer is going to be a link to any website I can assure you I read them all and that only left me more confused. So here are my questions: Homework Equations 3. The Attempt at a Solution [/B] ->...

For a series to be convergent,it must have a finite sum,i.e.,limiting value of sum.As the sum of n terms approaches a limit,it means that the nth term is getting smaller and tending to 0,but why is not the converse true?Should not the sum approach a finite value if the nth term of the series is...

Hello, I was given f(-4x)= 1/(1+4x), and I used the geometric series to find the power series representation of this function. I then took the limit of (-4x)^k by using ratio test. The answer is abs. value of x. So -1/4<x<1/4 I then plugged in those end points to the series going from k=0 to...

Is there a simple closed-form solution for the following infinite series? ##F(a,b,c) = \sum_{j=0}^\infty \frac{(j+a)!}{(j+b)! (j+c)!}## where ##a, b, c## are positive integers?

Homework Statement The Attempt at a Solution I have deliberately made several obvious steps, because I keep ending up here. However I have no idea what to do from here. I thought about the fact, that differential equations have the solution ##x = x_{HOM} + x_{Inhom}##, but the ##x_{HOM}##...

Hi guys! Here's a problem i was working on. I solved it by root test and got absolute value of x on the outside of the limit and the limit equaled zero. Is it wrong to multiply the outside absolute value by the zero I got from the limit? or is that okay? In general, when we are solving power...

Homework Statement Use an appropriate Laurent series to find the indicated residue for ##f(z)=\frac{4z-6}{z(2-z)}## ; ##\operatorname{Res}(f(z),0)## Homework Equations n/a The Attempt at a Solution Computations are done such that ##0 \lt \vert z\vert \lt 2##...

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...