Series Definition and 998 Threads

Proof: Let ## \epsilon=0 ##. Then the unperturbed equation is ## \ddot{x}+x=0 ## and the general solution is ## x(t)=A\sin\omega t+B\cos\omega t ## where ## \omega=1 ## is the angular frequency with the constants ## A ## and ## B ##. With the initial condition ## x(0)=0 ##, we have that ##...

The question was: calculate the following sum within its open interval of convergence after determining the radius of convergence: ##\sum_{n=0}^{+\infty} \frac{x^n}{2n+1}## ##\textbf{Finding the radius of convergence:}## I believe I followed the correct steps, but I got stuck solving it...

is there an equivalent to taylor series but with differences instead of derivatives ? are Newton series analogue to Taylor series ?

Mentor note: Moved from technical math section, so missing the homework template filling in the boxes is easy, 1,3,6,10,15 second is just squares of that, 1,9,36,100,225 But how I anyone supposed to find and expression for this? This is from a textbook on elementary algebra, the specific...

I cannot understand why the 45 ohm resistor is parallel to the 15 ohm resistor. It's been defined that if two or more resistors are in parallel they same common connection points. I don't see these connections points. Is the junction point after the ammeter signifying that the 45 ohm and 15 ohm...

I was intrigued by a comment in Brilliant.org: Besides the proof provided by Brilliant, I also found a couple of other websites. But none of these proofs were entirely clear to me. So I tried to come up with my own proof. Since I am not a group theorist, I wanted to ask if the proof makes...

I have another dilemma with terminology that is puzzling and would appreciate some advice. Consider the following truncated Taylor Series: $$\begin{equation*} f(\vec{z}_{k+1}) \approx f(\vec{z}_k) + \frac{\partial f(\vec{z}_k)}{\partial x} \Delta x + \frac{\partial f(\vec{z}_k)}{\partial...

Greetings, I would like to gain some insight when it comes to dealing with this problem. Personally, I wasn't able to solve it. I had to look for the solution in the book. I can just tell you that it's a telescopic series, the rest would be too much. You don't have to post the whole solution...

We are working with series currently, and some of the problems ask you to create a general term and write it in series summation form. Some of it is a no-brainer, but other ones, I'm at a loss as to how they expect us to get the answer without a ton of trial and error. For example, there is...

I have tried a few things and can't figure this out. If I separate the top and bottom, the top obviously quickly goes to 0, so there's no series after 1 derivative (f'(2x) = 2, then f''(2x) = 0), so I can't separate them and do anything with it And if I do it with ## 2x(e^{2x}-1)^{-1} ##...

For context, this is when deriving the Boltzmann distribution by using a canonical ensemble (thermodynamics). omega is a function to represent number of microstates. According to wikipedia... is the first order expansion around 0 (Maclaurin series). My confusion: What are even...

As far as how far I've gotten, I split the non-repeating portion of the series apart from the repeating portion, set r as ## 10^{-6} ## and get this: ## 0.65+285714/9999990 ## From here though, I don't see how to simplify that fraction without something extremely tedious, like pulling out...

Hi all, I came across the following stackexchange post and was wondering if anyone could give any elaboration for why the answer claims that evaluating the Taylor Series resulted in ##\mathcal{O}(\epsilon^{3})## errors? I have not encountered such an expansion before. EDIT: The equation at hand...

For this problem, Let ##a_n = \frac{1}{n(\ln n)^p}## ##b_n = \frac{1}{(n \ln n)^p} = \frac{1}{(n^*)^p}## We know that ##\sum_{2 \ln 2}^{\infty} \frac{1}{(n^*)^p}## is a p-series with ##n^* = n\ln n##, ##n^* \in \mathbf{R}## Assume p-series stilll has the same property when ##n^* \in...

So here you can see the basics behind a thermocouple. What people usually do, is that they connect these junctions electrically in series and thermally in parallel. Now another thing people do is cascading several peltier modules. Now my thought was, why not connect them electrically...

1. ##f(z)=\dfrac {\sin z}{z- \pi}## at ##z=\pi## : $$ \dfrac {\sin z}{z- \pi}=\dfrac {\sin(\pi +z- \pi)}{z- \pi}=\dfrac {- \sin(z- \pi)}{z- \pi}=\dfrac {-1}{z- \pi} \sum_{n=0}^\infty \dfrac {(-1)^n (z- \pi)^{2n}}{(2n+1)!}$$My answer has extra ##\dfrac {-1}{z- \pi} ## according to a calculator...

Hello, this is my attempt for #19 for 11.6 of Stewart's “Multivariable Calculus”. The question is to determine whether the series is absolutely convergent, conditionally convergent, or divergent. The answer solutions used a ratio test to reach the same conclusion but I used the comparison test...

Find, with proof, the sum of the alternating series $$\sum_{n = 0}^\infty \frac{(-1)^n}{(2n+1)^3}$$

My interest is on the (highlighted part in yellow ) of finding the partial fractions- Phew took me time to figure out this out :cool: My approach on the highlighted part; i let ##(kr+1) =x ## then, ##\dfrac{1}{(kr+1)(kr-k+1)} = \dfrac{1}{x(x-k)}## then...

Is the infinite series ##\sum_{n=1,3,5,...}^\infty \frac {1} {n^6}## somewhat related to the Riemann zeta function?The attached image suggest the value to be inverse of the co-efficient of the series.Is there any integral representation of the series from where the series can be evaluated?

I'm stuck on a problem: T1 = dT2/dt + xT2 - y T2 = (Ae^(-4.26t))+(Be^(-1.82t))+39.9 I'm unsure how to proceed

Any set of a series of numbers consisting of increasing integer members, all of which are determined by a common proposition or characteristic, will always be infinite in size. Examples… Prime numbers Mersenne primes Odd perfect numbers(if they exist) Zeroes of the Zeta function Regardless...

Consider the above diagram. Once the first capacitor is charged, clearly it will have a voltage ##E##. Then when the switch is flipped, the cell no longer matters (there is no complete circuit which goes through the cell), so we have the first capacitor connected to the second one, and it looks...

By definition: ##\log_e(1+x)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}- \cdots ## ##(1)## Replacing ##x## by ##−x##, we have: ##\log_e(1-x)=-x-\dfrac{x^2}{2}-\dfrac{x^3}{3}- \cdots## By subtraction, ##\log_e(\dfrac{1+x}{1-x})=2(x+\dfrac{x^3}{3}+\dfrac{x^5}{5}+ \cdots)## Put ##...

So, the function is piecewise continuous (and differentiable), with (generalized) one-sided derivatives existing at the points of discontinuity. Hence I conclude from the theorem that the series converges pointwise for all ##t## to the function ##f##. I've double checked with WolframAlpha that...

The series that is given is $$\frac12+\frac13+\left(\frac12\right)^2+\left(\frac13\right)^2+\left(\frac12\right)^3+\left(\frac13\right)^3+\ldots.$$ Now, it's easy to see these are two separate geometric series, however, Spivak claims the ratio test fails because the ratio of successive terms...

Hi, I am having problems with task d) I now wanted to check the convergence using the quotient test, so ## \lim_{n\to\infty} |\frac{a_{n+1}}{a_n}| < 1## I have now proceeded as follows: ##\frac{a_{n+1}}{a_n}=\frac{\Bigl( 1 + \frac{1}{k+1} \Bigr)^{(k+1)^2}}{3^{k+1}} \cdot...

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I found how to get the solution to this question (the answer is 200V), but I don't understand why we ignore the 30kOhm resistor when using analysing the circuit. Because it is in series with the open voltage, wouldn't there be some voltage drop across the resistor that would affect the...

Prove that the series $$\sum_{k = 1}^\infty \frac{(-1)^{k-1}}{|x| + k}$$ converges for all ##x\in \mathbb{R}## to a Lipschitz function on ##\mathbb{R}##.

I found the answer for the springs in parallel, but not for the ones in series. I believe I don't understand how the forces are interacting properly. Here's a force diagram I drew. Everytime I try to make equations from this though my answer dosen't make sense. The mass m has a gravititoanl...

Show that $$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$$

TL;DR Summary: I am looking for a good thorough book that is devoted to assembling and explaining techniques of evaluating series. evaluating series is a very big problem for me right now. I know nowhere near as much about it as I do integration, and the main reason for this is that its quite...

I've been working on developing infinitesimal recursion (what I call continuous hierarchy), but I ended up arriving at "field series" instead. My searches didn't seem to come up with anything reasonable (battlefield the video game series), so I'm wondering what the official name for a field...

So my idea was to separate the capacitor into two individual ones, one of length ##l - a## filled with a vacuum and one of length ##a## filled with the glass tube. The capacitances then are $$ C_0 = \frac{2 \pi \varepsilon_0 (l-a)}{\displaystyle \ln\left( \frac{r_2}{r_1} \right)} $$ for the...

Margules suggested a power series formula for expressing the activity composition variation of a binary system. lnγ1=α1x2+(1/2)α2x2^2+(1/3)α3x2^3+... lnγ2=β1x1+(1/2)β2x1^2+(1/3)β3x1^3+... Applying the Gibbs-Duhem equation with ignoring coefficients αi's and βi's higher than i=3, we can obtain...

Let ##\{a_n\}_{n = 1}^\infty## be a sequence of real numbers such that for some real number ##p > 1##, ##\frac{a_n}{a_{n+1}} = 1 + \frac{p}{n} + b_n## where ##\sum b_n## converges absolutely. Show that ##\sum a_n## also converges absolutely.

Evaluate the Fourier series $$\frac{1}{\pi^2}\sum_{k = 1}^\infty \frac{\cos 2\pi kx}{k^2}$$ for ##0 \le x \le 1##.

This is an Argand diagram showing the first 40,000 terms of the series form of the Riemann Zeta function, for the argument ##\sigma + i t = 1/2 + 62854.13 \thinspace i## The blue lines are the first 100 (or so) terms, and the rest of the terms are in red. The plot shows a kind of approximate...

This is the series: $$\sum_{n=1}^{+\infty}\sin(n)\sin\left(\frac{1}{n}\right)\left(\cos\left(\frac{1}{\sqrt{n}}\right)-1\right)$$

TL;DR Summary: I want to find a function with f'>0, f''<0 and takes the values 2, 2^2, 2^3, 2^4,..., 2^n Hello everyone. A professor explained the St. Petersburgh paradox in class and the concept of utility function U used to explain why someone won't play a betting game with an infinite...

##\sum_{n=1}^\infty n^{-a}## converge s for ##a\gt 1## - otherwise diverges. Is there any theory for ##a_n##? For example ##a_n\gt 1## and ##\lim_{n\to \infty} a_n =1##. How about non-convergent with ##\liminf a_n=1##?

I had difficulty showing this no matter what I tried in (a) I am not getting it. Here for p(t) in K[[t]] : ## |p|=e^{-v(p)} ## where v(p) is the minimal index with a non-zero coiefficient. I said that p_i is a cauchy sequence so, for every epsilon>0 there exists a natural N such that for all...

Here I list my problem in the attachment.

Can you please explain this series f(x+\alpha)=\sum^{\infty}_{n=0}\frac{\alpha^n}{n!}\frac{d^nf}{dx^n} I am confused. Around which point is this Taylor series?

My question is; is showing the highlighted step necessary? given the fact that ##\sin (nπ)=0##? My question is in general i.e when solving such questions do i have to bother with showing the highlighted part... secondly, Can i have ##f(x)## in place of ##x^2##? Generally, on problems to do...

Attempt; ##\dfrac{1}{r(r+1)(r+2)} -\dfrac{1}{(r+1)(r+2)(r+3)}=\dfrac{(r+3)-1(r)}{r(r+1)(r+2)(r+3)}=\dfrac{3}{r(r+1)(r+2)(r+3)}## Let ##f(r)=\dfrac{1}{r(r+1)(r+2)}## ##f(r+1)= \dfrac{1}{(r+1)(r+2)(r+3)}## Therefore ##\dfrac{3}{r(r+1)(r+2)(r+3)}## is of the form ##f(r)-f(r+1)## When...

My attempt; ##r^2+r-r^2+r=2r## Let ##f(r)=(r-1)r## then it follows that ##f(r+1)=r(r+1)## so that ##2r## is of the form ##f(r+1)-f(r)##. When ##r=1;## ##[2×1]=2-0## ##r=2;## ##[2×2]=6-2## ##r=3;## ##[2×3]=12-6## ##r=4;## ##[2×4]=20-12## ... ##r=n-1##, We shall have...

We're off grid at 57degrees north. Our source of electricity is solar panels, with a diesel generator as backup. The solar has served us well, until this November, where we had almost 8 weeks of 0 sunshine. I got sick of running the generator. It's noisy, it needs refueling, smells... So I...

Hi can someone tell me please how much current is passed though below circuit: 120 AC 60Hz mains power going through a 10uF 500v capacitor in series