Series Definition and 998 Threads

I've recently purchased Schaum's Elementary, Intermediate and College Algebra texts and I'm loving the extra practice and the chance to familiarize myself with the material. Are there any other similar books out there for cheap? I've considering buying old editions of the various...

good day I want to study the convergence of this serie and want to check my approch I want to procede by asymptotic comparison artgln n ≈pi/2 n+n ln^2 n ≈n ln^2 n and we know that 1/(n ln^2 n ) converge so the initial serie converge many thanks in advance!

Good day I want to study the connvergence of this serie I already have the solution but I want to discuss my approach and get your opinion about it it s clear that n^2+5n+7>n^2+3n+1 so 0<(n^2+3n+1)/(n^2+5n+7)<1 so we can consider this as a geometric serie that converge? many thanks in advance

Good day and here is the solution, I have questions about I don't understand why when in the taylor expansion of exponential when x goes to infinity x^7 is little o of x ? I could undesrtand if -1<x<1 but not if x tends to infinity? many thanks in advance!

Good day here is the exercice and here is the solution that I understand very well but I have a confusion I hope someone can explain me if I take the taylor expansion of sin ((n^2+n+1/(n+1))*pi)≈n^2+n+1/(n+1))*pi≈n*pi which diverge! I know something is wrong in my logic please help me many...

Hi, I am not sure what the correct forum is for this question. Question: When do we need to remove seasonality from time series data to do a regression analysis? Context: I am planning to conduct a prediction analysis where I want to find out how a device performs. I hope to estimate a...

Hello Forum, Some of my electrical outlets (3) in the kitchen stopped working (one of them is a GFCI outlet). Reading online, I found out that outlets are generally connect in a daisy-chain fashion and if one goes back they all stop working. See the figure below showing a daisy chain...

i looked at this and it was not making any sense at all, could it be a textbook error or i am missing something here; note that, lhs gives us, ##4,6,8,10,12,14## rhs gives us, ##8,11,14,17,20,23##

The answer in the textbook writes: $$ f(x) = \frac{1}{4} +\frac{1}{\pi}(\frac{\cos(x)}{1}-\frac{\cos(3x)}{3}+\frac{\cos(5x)}{5} \dots) + \frac{1}{\pi}(\frac{\sin(x)}{1}-\frac{2\sin(2x)}{2}+\frac{\sin(3x)}{3} + \frac{\sin(5x)}{5}\dots)$$ I am ok with the two trigonometric series in the answer...

Hi, I have a question about a homework problem: I am not sure why I do not seem to get the same answers when using different methods. Question: Given transfer functions G(s) = \frac{s - 1}{s + 4} and C(s) = \frac{1}{s - 1} , find the state space models for those systems. Then find the...

So, you know that thing with the Wizard of Oz you can't unsee once someone explains how it's actually Glenda the Good Witch who's the real villain? (Glenda blatantly endangered and manipulated Dorothy into killing Glenda's most dangerous enemy, while smiling. Ending up with the Ruby Slippers...

Can anyone suggest a good lecture series on Complex Analysis on YouTube? I have already searched on YouTube myself, and there are a few. But I wanted to know if any of you would recommend some particular lecture series which you consider to be good.

I have a seen a number of great series about women lately. The Queens Gambit: Loved it! The Marvelous Mrs. Maisel: Of course this has been around since 2013 and just concluded last year Orange is the New Black All on Netflix

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The Alternating series test has to be used to determine whether this series converges or diverges: \sum\limits_{n=1}^{\infty} (-1)^n\frac{\sqrt n}{2n+3} Here's what I have done: Let a_n = \frac{\sqrt n}{2n+3}. Therefore, a_{n+1} = \frac{\sqrt {n+1}}{2n+5} Now, for a_{n+1} to be less than or...

Here's my attempt: clc; clear all; Ra=1.8; La=150*10^(-3); Ls=150*10^(-3); Rs=1.8; Rt=Rs+Ra; ws=100*pi; c=1.25; vrms=240; vm=240*sqrt(2); f=50; T=1:0.5:50; for a=0:pi/9:pi/3 vt=(2*vm*cos(a))/pi wc=((ws*(Ls+La)/tan(a))-Rs-Ra)/c; x=[wc; a+pi]; Tc=c*(2*vm*cos(a)/(pi*(c*wc+Ra+Rs)))^2; for...

First i computed the rate of energy wrt time of the second plate: $$dq_{2}/dt = A \sigma ((373)^4/2 + (273)^4/2 - T_{2}^4)$$ Equaliting it to zero we get the answer. But i am not sure why do we need to equality it to zero. The q arrow on the figure suggest me that it is conservation of energy...

The spectrochemical series of metals, under the circumstances that same ligands are used and that it is in an octahedral coordination, is given by: Mn2+ < Ni2+ < Co2+ < Fe2+ < V2+ < Fe3+ < V3+ < Co3+ < Mn4+ < Mo3+ < Rh3+ < Ru3+ < Pd4+ < Ir3+ < Pt4+ When I was skimming through a textbook to...

Attached is the section from the book. I am doing section 31.3 We know that an AC source gives a sinusoidal varying current, and as far as I know its always given by ##i(t) = Icos(wt)##. Its like we take the current to be the base of all other quantities, so we use it to derive everything else...

My attempt: If ##i = 1##, then ##\gamma_1 = G \rhd G' = \gamma_2##. We proceed by induction on ##i##. Consider an element ##xyx^{-1}y^{-1}## where ##x \in \gamma_i## and ##y \in G##. Since ##\gamma_i \rhd G##, we have ##yx^{-1}y^{-1} = x_0 \in \gamma_i##. So, ##xyx^{-1}y^{-1} = xx_0 \in \gamma_i...

Hello, I am currently self studying sequence and series and I got to a topic called arithmetic-geometric sequence, and after the theory It gives this exercise: 1) Find the sum: S=1+11+111+1111+...+111...111, if the last (number) is a digit of n. I was given a tip That says that 1 = (10 - 1)/9...

I am lookin designing balancing resistors for series capacitors and understand that I need to consider the leakage current from the capacitors. I am trying to determine factors that would case the insulation resistance to decrease over time so I can design around that.

Very simply, I can't understand why the charges of capacitors placed in series are all the same, and why even the total one(of the circuit) is equal to those. How is it possible that the total charge is the same as the individual ones? There must be some concept/property about capacitors which...

Some popular math videos point out that, for example, the value of -1/12 for the divergent sum 1 + 2 + 3 + 4 ... can be found by integrating n/2(n+1) from -1 to 0. We can easily verify a similar result for the sum of k^2, k^3 and so on. Is there an elementary way to connect this with the more...

If given a set of data points for the magnitude of a cepheid variable at a certain time (JD), how can we use Fourier series to find the period of the cepheid variable? I'm trying to do a math investigation (IB math investigation) on finding the period of the cepheid variable M31_V1 from data...

Mary Boas attempts to explain this by pointing out that the situation cannot arise because charges will have to be placed individually, and in an order, and that order would represent the order we sum in. That at any point the unplaced infinite charges would form an infinite divergent series...

Attempt: Consider an arbitrary normal series ##G = G_0 \ge G_1 \ge G_2 \ge \dots \ge G_n = 1##. We will refine this series into a composition series. We start by adding maximal normal subgroups in between ##G_0## and ##G_1##. If ##G_0/G_1## is simple, then we don't have to do anything. Choose...

Attempt so far: We're given that ##G## and ##H## have equivalent normal series $$G = G_0 \ge G_1 \ge \dots \ge G_n = 1$$ and $$H = H_0 \ge H_1 \ge \dots \ge H_n = 1$$ We can assume they have the same length because they are equivalent. I think from here I need to construct two composition...

So here's my homework question: This is the reference formula along with the Rung-Kutta form with the variables mentioned in the question Here is my attempt so far: Problem is that i am unsure how to expand this to even get going. I tried referencing my text Math Methods by Boas which has...

Prove $$ \zeta(2) = \sum_{n\in \mathbb{N}}\dfrac{1}{n^2} = \dfrac{\pi^2}{6} $$ by evaluating $$ \int_0^1\int_0^1\dfrac{1}{1-xy}\,dx\,dy $$ twice: via the geometric series and via the substitutions ##u=\dfrac{y+x}{2}\, , \,v=\dfrac{y-x}{2}##.

I'm trying to get from the formula in the top to the formula in the bottom (See image: Series). My approach was to complexify the sine term and then use the fact that (see image: Series 1) for the infinite sum of 1/ne^-n. Then use the identity (see image: Series 2). Any other ideas?

Why we say that the resistance of the series curcuit is equal to the sum of the resistances of the resistors?

Would somebody be kind enough to check whether I've picked the right convergence tests for each of these and reached the right answers? There are no solutions in the book. Also, is there a method I can use to determine if I'm right - does calculating the first n terms help? Thank you Edit...

If the current goes through the first end of the resistor will it be less than on the other end of resistor?

Prove that $\sqrt[3]{\dfrac{2}{1}}+\sqrt[3]{\dfrac{3}{2}}+\cdots+\sqrt[3]{\dfrac{996}{995}}-\dfrac{1989}{2}<\dfrac{1}{3}+\dfrac{1}{6}+\cdots+\dfrac{1}{8961}$.

hi guys i am trying to solve the Gaussian integral using the power series , and i am suck at some point : the idea was to use the following series : $$\lim_{x→∞}\sum_{n=0}^∞ \frac{(-1)^{n}}{2n+1}\;x^{2n+1} = \frac{\pi}{2}$$ to evaluate the Gaussian integral as its series some how slimier ...

I was trying to find this form of the Taylor series online: $$\vec f(\vec x+\vec a) = \sum_{n=0}^{\infty}\frac{1}{n!}(\vec a \cdot \nabla)^n\vec f(\vec x)$$ But I can’t find it anywhere. Can someone confirm it’s validity and/or provide any links which mention it? It seems quite powerful to be...

I have the following function $$f^{(0)}\left(x\right)=f\left(x\right)=e^{x}$$ And want to approximate it using Taylor at the point ##\frac{1}{\sqrt e} ## I also want to decide (without calculator)whether the error in the approximation is smaller than ##\frac{1}{25} ## The Taylor polynomial is...

Hi! Some time ago I came across a series and never solved it, I tried to give a new go because I was genuinely curious how to tackle it, which I thought would work, because it looks innocent, but there is something about the beast making it hard to approach for me. So need some help! Maybe this...

Given the circuit above, I have to solve for the labelled currents, find V total and R total accordingly. 1A is flowing through the 5Ω resistor as shown. Assuming electron flow (negative terminal to positive) for circuit. The connector in the middle was somewhat confusing. Without it, this...

Can someone help me on this question? I'm finding a very strange probability distribution. Question: Suppose that x_1 and x_2 are independent with x_1 ~ geometric(p) and x_2 ~ geometric (1-p). That's x_1 has geometric distribution with parameter p and x_2 has geometric distribution with...

My homework is on mathematical physics and I want to know the concept behind Laurent series. I want to know clearly know the process behind attaining the series representation for the expansion in sigma notation using the formula that can be found on the attached files. There are three questions...

Question 1; Method 1 If the sum of the first four terms is 139 then S4=139 139=1/2(4)(2a+(4-1)d) 139=2(2a+3d) 139=4a+6d----- [1] The part of this question that is confusing is the "the sum of the next four terms is 115". Would this mean that S8=S4+115=139+115=254? In which case...

A beam of length L with fixed ends, has a concentrated force P applied in the center exactly in L / 2. In the differential equation: \[ \frac{d^4y(x)}{dx^4}=\frac{1}{\text{EI}}q(x) \] In which \[ q(x)= P \delta(x-\frac{L}{2}) \] P represents an infinitely concentrated charge distribution...

The American Physical Society (APS) has just launched a webinar series addressing careers for physicists in industry (broadly includes national labs). [ETA: Correction: National labs are covered under a separate series also listed on the following link.] Details of the series can be found...

So this seems to be a geometric Series, but with the coefficients in front, how do I exactly go about proving this? Thanks

Show by contradiction that $$ \sum_{p\in \mathbb{P}}\dfrac{1}{p} =\sum_{p\;\text{prime}}\dfrac{1}{p} $$ diverges. Which famous result is an immediate corollary?

The sum of the harmonic series(1/1+1/2+1/3...) is infinite. However, if you exclude all the terms that contain the number nine, the sum is just under 23. From 1 to 100 19% of the terms are excluded From 1 to 1000 27.1% of the terms are excluded Is there a formula for a N digit number what the...

Hi, Question: If we have an initial condition, valid for -L \leq x \leq L : f(x) = \frac{40x}{L} how can I utilise a know Fourier series to get to the solution without doing the integration (I know the integral isn't tricky, but still this method might help out in other situations)? We are...

Hey! 😊 I want to check the convergence for the below series. - $\displaystyle{\sum_{n=1}^{+\infty}\frac{\left (n!\right )^2}{\left (2n+1\right )!}4^n}$ Let $\displaystyle{a_n=\frac{\left (n!\right )^2}{\left (2n+1\right )!}\cdot 4^n}$. Then we have that \begin{align*}a_{n+1}&=\frac{\left...