Series Definition and 998 Threads

Sorry if the answer is obvious, but I was wondering if positive and negative electrodes (cells in series) can share the same current collector as depicted below? I want to create a 12V battery with cells inline in series without creating cells with individual current collectors. Note that the...

##\sum_{k=0}^\infty \frac {2^n+3^n}{4^n+5^n} x^n## in order to find the radius of convergence i apply the root test, that is ##\lim_{n \rightarrow +\infty} \sqrt [n]\frac {2^n+3^n}{4^n+5^n}## ##\lim_{n \rightarrow +\infty} \left(\frac {2^n+3^n}{4^n+5^n}\right)^\left(\frac 1 n\right)=\lim_{n...

given the following ##\sum_{n=0}^\infty n^2 x^n## in order to find the radius of convergence i do as follows ##\lim_{n \rightarrow +\infty} \left |\sqrt [n]{n^2}\right|=1## hence the radius of convergence is R=##\frac 1 1=1## |x|<1 Now i have to verify how the series behaves at the...

##\sum_{n=0}^\infty (-1)^n \frac {x^\left(n+1\right)}{n+1}## for x=1 ##\sum_{n=0}^\infty (-1)^n \frac {1^\left(n+1\right)}{n+1}## i've tried leibniz test but i can only find that it converges why is this power equal to ##log(2)##? i've also tried with ##\sum_{n=0}^\infty\log \left (1+\frac 1...

ANY AND ALL HELP IS GREATLY APPRECIATED :smile: I have found old posts for this question however after reading through them several times I am having a hard time knowing where to start. I am happy with the sketch that the function is correctly drawn and is neither odd nor even. It's title is...

##\sum_{n=1}^\infty \frac {(sin 𝜶)^n}{2n} ## I apply the root test and i get ##\lim_{n \rightarrow +\infty} \frac {sin 𝜶}{2n^\frac 1 n} ## at this point i don't know how to treat the denominator.

## \sum_{n=1}^\infty (-1)^n \frac {log(n)}{e^n}## i take the absolute value and consider just ## \frac {log(n)}{e^n}## i check by computing the limit if the necessary condition for convergence is satisfied ##\lim_{n \rightarrow +\infty} \frac {log(n)}{e^n} =\lim_{n \rightarrow +\infty}...

## \sum_{n=0}^\infty \frac {(2n)!}{(n!)^2} ## ##\lim_{n \rightarrow +\infty} {\frac {a_{n+1}} {a_n}}## that becomes ##\lim_{n \rightarrow +\infty} {\frac { \frac {(2(n+1))!}{((n+1)!)^2}} { \frac {(2n)!}{(n!)^2}}}## ##\lim_{n \rightarrow +\infty} \frac {(2(n+1))!(n!)^2}{((n+1)!)^2(2n)!}##...

This is written on Greiner's Classical Mechanics when solving a Tautochrone problem. Firstly,I don’t understand why we didn’t use the term ##m=0## and Sencondly, how the integrand helps us to fulfill the Dirichlet conditions. That means,how do we know that the period is 1?Thanks

I find this interesting. A pretty detailed description, of a complex geological series of events, that can't be directly seen. Here's my summary: In 2018 an usual humming was picked up by seismic equipment an island off Africa, a magma pool drained, flowed up a dyke, when horizontal, and then...

I found the Req which is 13.6 and also found the It which is 0.74. I'm having trouble finding the separate current and potential difference numbers.

I'll make a power bank with capacitors and I made circuit of it. But I'm worrying about whether the circuit is safe, because it's dangerous to use capacitor. So, can you check the circuit i made?? My capacitor is 2.7V, 600F and the power bank circuit has "Charging current : 1A maximum, output...

I found that ρn = √(2n+1)/(n+1). Then, I found ρ = lim when n→∞ |(1/n) (√(2n+1))/((1/n) (n+1))| = 0 Based on this result I concluded the series converges; however, the book answer says it diverges. What am I doing wrong?

Can someone help me understand why what I wrote is correct? That is: If I have a sequence with double indices and if the summation of the elements modules of this sequence converges (less than infinite) than it does not matter how I make this sum (second line) they are going to be always the...

Hello, Hello, For a project , i need to modele a photodiode with a current source in paralelle with a shunt resistance and in serie with a resistance to use it in a bigger circuit. The photodiode we will use is SFH7050, the datashhet is provideed here...

Consider the following series with the following pattern $$\frac {1}{1×3}+\frac {1}{5×7}+\frac {1}{9×11}...$$ How would you go about working out what the general rule for this sum is? That is in the form of ##\sum_{n=a}^{b}f(n)## Any help is greatly appreciated.

I got answer to (a), which is 3/4 sin thteta - sin ((3^(n+1)) theta) / (4 . 3^n) but I do not know how to use this result to prove next question. I tried to change theta into pi/2 - theta so that sin change to cos or vice versa but not working. Thanks

I know i have to use the efficiency formula and everything is fine but i don't know how to find T its the only unknown in my equation can someone please tell me how to find T . In the solution they got the value of T by equating the work done by the two engines , but why is their work done equal ?

I have an ODE: (x-1)y'' + (3x-1)y' + y = 0 I need to find the solution about x=0. Since this is an ordinary point, I can use the regular power series solution. Let y = ## \sum_{r=0}^\infty a_r x^r ## after finding the derivatives and putting in the ODE, I have: ## \sum_{r=0}^\infty a_r...

After evaluating the integral I found the following: (1/3)tan-1(e∞/3) = (1/3)tan-1(∞) = (1/3)(nπ/2), where n is an odd number. In this case I found multiple solutions to the problem. How do you prove it converges?

I already found ##I(t)## using Kirchhoff's laws, I got the equation ##V-RI-L\frac{dI}{dt}=0\Rightarrow L\frac{dI}{dt}=V-RI## then I solved the differential equation getting ##I(t)=\frac{V}{R}\left[1-e^{-\frac{R}{L}t}\right]##. My problem is founding the voltage as a function of time ##V(t)##, I...

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}e^{-n}\sum^{n}_{k=0}\frac{n^k}{k!}$

I got the following expression: -(1/4)ln((n+2)/(n-2)) When I substitute "∞" in the expression I found it undefined. However, the book says the series converges. What am I doing wrong?

Solve the boundary value problem Given u_{t}=u_{xx} u(0, t) = u(\pi ,t)=0 u(x, 0) = f(x) f(x)=\left\{\begin{matrix} x; 0 < x < \frac{\pi}{2}\\ \pi-x; \frac{\pi}{2} < x < \pi \end{matrix}\right. L is π - 0=π λ = α2 since 0 and -α lead to trivial solutions Let u = XT X{T}'={X}''T...

Hi, this is my try:Thanks.

I'm having a hard time understanding how exactly to evaluate the expression} $$\partial_t \mathcal{T}\exp\left(-i S(t)\right)\quad \text{where}\quad S(t)\equiv\int_{t_0}^tdu \,H(u) .$$ The confusing part for me is that if we can consider the following: $$\partial_t \mathcal{T}\exp\left(-i...

converge or diverge $$S_n= \sum_{n=1}^{\infty} (-1)^{n+1}\frac{\sqrt{n}+6}{n+4}$$ ok by graph the first 10 terms it looks alterations are converging to 0

How i can find r2 value in this circuit? R1 = 2 R3 = 5 E = 20v

In order to obtain equation (3), I think I have to do the Fourier transform in the x direction: \begin{equation} \tilde{G}(k,y,x_0,y_0) = \int_{- \infty}^{\infty} G(x,y,x_0,y_0) e^{-i k x} dx \end{equation} So I have: \begin{equation} -k ^2 \tilde{G}(k,y,x_0,y_0) + \frac{\partial^2...

Greg has kindly allowed me to post these equations which I compiled many years ago. Somehow I like them better than anything I've ever run across so maybe someone else will find them useful also. Actually, I have given some thought to the Fourier series and how they tie in with sampled-data...

In this circuit a battery,Capacitor,and a resistance are in series. For simplicity assume that there is a +4V in the positive terminal of the battery and -4V in the negative one and let A be the capacitor plate connected to the positive terminal and B the capacitor plate connected to the...

(In opening, hi. I'm a lawyer, not a physicist, and I'm entirely out of my depth here.) I need to make roughly 8 wooden frames (19 inches x 1-1/16 inches x 9-1/8 inches). Into each, I'd like to place a thin metal plate with hexagonal cells (5.27 mm cell diameter) pressed into them. I need to...

Use the comparison test to determine if the series series convergences or divergences $$S_{6}=\sum_{n=1}^{\infty} \dfrac{1}{n^2 \ln{n} -10}$$ ok if i follow the example given the next step alegedly would be... $$\dfrac{1}{n^2 \ln{n} -10}<\dfrac{1}{n^2 \ln{n}}$$ $\tiny{242 UHM}$

Hi, I have a particular equation in a paper, wherein the author specifies an infinite series. The author has apparently found the sum of the series and calculated the equation. Can anyone please help me in understanding how to sum such a series. I have attached part of the paper with the...

I'm using the sum of a geometric series formula, but I'm not sure how to find the ratio, r. The n is confusing me. The solution is below, but I'm having trouble with the penultimate step.

I'm not really sure what I need to find exactly. From what I'm seeing, I could give C1 the max potential difference of 125V because it has the lowest capacitance, and because V = Q/C, this means the capacitor with the highest potential difference across its plates will be the one with the lowest...

Hi, I'm trying to solve the sum of following infinite series: $$ \sum_{k=1}^{\infty} \frac{{k}^{2}+4}{{2}^{k}} = \sum_{k=1}^{\infty} \frac{{k}^{2}}{{2}^{k}} + \sum_{k=1}^{\infty} \frac{4}{{2}^{k}}$$ Using partial sum we can rewrite the first series: $$ \sum_{k=1}^{\infty}...

To anyone that can help me with this - You have to pick the FIRST correct reason. Work below (exception of 4 because I cannot figure it out), but in order to get the question right you must have all correct and I cannot figure it out. Any help is appreciated. [Moderator's note: Moved from a...

I know that in parallel springs, x (the displacement of the spring) is the same for both springs. However, the forces resulting for each string are different. For springs in a series, x may be different, but the force is the same on each string. I got the answer b, seeing how the weight would...

Summary: Can someone give me a basic high level overview on how to do a binomial expansion? I'm studying for my E&M test and going over multipole expansion. I'm particularly confused about these lines (Griffiths E&M 4th Edition) 𝓇^2_{\pm} = r^2 \left(1\mp \frac{d}{r} \cos\theta +...

I tried by ##S=1+(1/1!)(1/4)+(1.3/2!)(1/4)^2+...## ##S/4=1/4+(1/1!)(1/4)^2+(1.3/2!)(1/4)^3..## And then subtracting the two equations but i arrived at nothing What shall i do further?

Hello, I'm using a "TQ2SA-1.5V Panasonic 2 Form C AS Single side stable, 1.5VDC 2A DPDT NON-LATCHING SMD Relay" (specifically the coil side of this relay) that is rated for 1.5 volts that is connected in series to this circuit (as the last device in this circuit shown below), which in this...

No idea on this one - I know that the spring constant will divide by 3 but am unsure how this will affect the % and the absolute uncertainties. Completely stuck on the extension...

Can somebody explain this please? I don't understand this.

I have no problems with part a). I used the formula for capacitance and determined the charges to be 0.00125 coulombs and 0.002 coulombs. The solution in the book is the same. For part b) my initial thought was that the charges will redistribute themselves so that each capacitor get the same...

I have taken a look but most books and Online stuff just menctions the First order Taylor for 3 variables or the 2nd order Taylor series for just 2 variables. Could you please tell me which is the general expression for 2nd order Taylor series in 3 or more variables? Because I have not found...

i am planning to measure the back emf produced by inductor when you open a switch. i know it is very hard to predict the voltage. but is there any way to narrow the possibilities?

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I am reading an interesting book by Julian Havil called:" Gamma-Exploring Euler's Constant." Much of the book is devoted to the harmonic series,a slowly diverging series that tends toward infinity.However,one paragraph puzzles me. On p. 23 he says: " In 1968 John W. Wrench Jr calculated the...