Series Definition and 998 Threads

Hi - an example in my book shows that FS coefficiants can be arrived at by minimizing the integrated square of the deviation, i.e. $ \Delta_p = \int_0^{2\pi}\left[ f(x) - \frac{a_0}{2}-\sum_{n=1}^{p}\left( a_nCosnx + b_nSinnx \right) \right]^2dx $ So we're looking for $ \pd{\Delta_p}{a_n}...

Need help. Determine the convergence of the series: 1. sum (Sigma E) from n=1 to infinity of: 1/((2*n+3)*(ln(n+9))^2)) 2. sum (Sigma E) from n=1 to infinity of: arccos(1/(n^2+3)) I think the d'alembert is unlikely to help here.

Hi folks, Just looking for an explanation on capacitor principles. My understanding: A capacitor is made from two conductors ( which have the ability to hold charge) separated by an insulator. Therefore current cannot flow between the+ and - plates. Unless unwanted breakdown from excessive...

Homework Statement How Can i do this on matlap the question in Attached files Homework Equations The Attempt at a Solution i try a lot but i failed

Hi there, I am reading through a thesis and the author takes the infinite series: \begin{equation} u(x,t)=u_0(x)+u_1(x)\cos(\sigma t - \phi_1(x)) + u_1'(x)\cos(\sigma' t - \phi'_1(x))+\ldots \end{equation} and by letting σr be the difference between the frequencies σ and σ' changes the above...

Homework Statement If we have a number sequence such that: a0, a1 are given, and every other element is given as ##a_n=\frac{(a_{n-1} + a_{n-2})}{2} then express an in terms of a0, a1 and n , and fin the limit of an Homework EquationsThe Attempt at a Solution If i try to express a3 in terms of...

Homework Statement Find the Fourier series defined in the interval (-π,π) and sketch its sum over several periods. i) f(x) = 0 (-π < x < 1/2π) f(x) = 1 (1/2π < x < π) 2. Homework Equations ao/2 + ∑(ancos(nx) + bnsin(nx)) a0= 1/π∫f(x)dx an = 1/π ∫f(x)cos(nx) dx bn = 1/π ∫f(x) sin(nx) The...

I was looking at the solution for problem 6 and I am confused on taking the derivatives of the function f(x)= cos^2 (x) I took the first derivative and did get the answer f^(1) (x)= 2(cos(x)) (-sin (x)), but how does that simplify to -sin (2x)? Is there some trig identity that I am not aware...

Hello,I've been reading my calculus book,and I can't tell the difference between a Taylor Series and a Taylor Polynomial.Is there really any difference? Thanks in advance

Hello,*please refer to the table above. I started from x(n)=x(n*Ts)=x(t)*delta(t-nTs), how can we have finite terms for discrete time F.S can anyone provide me a derivation or proof for Discrete F.S.?

If current is always the same in a series circuit then how is a transformer able to make the current smaller when it increases the voltage? is this just an exception since with the voltage being higher the same amount of power is being provided?

They ask for both $ \sum_{n=0}^{\infty} p^n Cos nx, also \: p^n Sin (nx) $ I'm thinking De Moivre so $$\sum_{n=0}^{\infty}p^n (e^{ix})^n = \sum_{n=0}^{\infty} p^n(Cos x + i Sin x)^n= \sum_{n=0}^{\infty} (pCos x + ip Sin x)^n$$ I also tried a geometric series with a=1, $r=pe^{ix}$ But...

Hi hi, So I worked on this problem and I know I probably made a mistake somewhere towards the end so I was hoping one of you would catch it for me. Thank you! Pasteboard — Uploaded Image Pasteboard — Uploaded Image

Hello, In finding a taylor series of a function using substitution, is it possible to use substitution for known taylor series of a function ,using different centers, and still get the same result. For example, if we have the function 1/(1+(x^2)/6) is it possible to use the taylor series of...

Homework Statement Periodic function P=3 f(t) = 0 if 0<t<1 1 if 1<t<2 0 if 2<t<3 a) Draw the graph of the function in the interval of [-3,6] b) Calculate the Fourier series of f(x) by calculating the coefficient. Homework EquationsThe Attempt at a Solution a) in attached...

Homework Statement Homework Equations The Attempt at a Solution Since P=2L, L=1 ? a_o = 1/2 [ ∫(from -1 to 0) -dx + ∫(from 0 to 1) dx ] = 1/2 [ (0-1) + (1-0) ] = 1/2(0) = 0 a_n = - ∫ (from -1 to 0) cosnπx dx + ∫ (from 0 to 1) cosnπx dx = 0 b_n = - ∫ (from -1 to 0) sinnπx dx...

Hi - my sometimes surprising set-book asks to show by series expansion, that $ \frac{1}{2}ln\frac{x+1}{x-1} =coth^{-1} (x) $ I get LHS = $ x+\frac{{x}^{3}}{3}+\frac{{x}^{5}}{5}+... $, which I think $= tanh^{-1} $ but I have found different expansions for the hyperbolic inverses, so I'd...

Hi, question asks to set upper and lower bounds on $$\sum_{n=1}^{1,000,000} \frac{1}{n}$$ assuming (a) the Euler-Mascheroni constant is known and (b) not known. $\gamma = \lim_{{n}\to{\infty\left( \sum_{m=1}^{n} \frac{1}{m} \right)}} = 0.57721566$ and I found (a) easily (14.39272...), but no...

Hi, question is - show that the following series is convergent: $ \sum_{s}^{} \frac{(2s-1)!}{(2s)!(2s+1)}$ Hint: Stirlings asymptotic formula - which I find is : $n! = \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n $ I can see how this formula would simplify - but can't see how it relates to the...

Sorry for the bad English , do not speak the language very well. I posted this to know if the statement or " hypothesis " is correct . thank you very much =D. First Image:https://gyazo.com/7248311481c1273491db7d3608a5c48e Second Image:https://gyazo.com/d8fc52d0c99e0094a6a6fa7d0e5273b6 Third...

The text does it thusly: imgur link: http://i.imgur.com/Xj2z1Cr.jpg But, before I got to here, I attempted it in a different way and want to know if it is still valid. Check that f^{*}f is finite, by checking that it converges. f^{*}f = a_0^2 + a_1^2 cos^2x + b_1^2sin^2x + a_2^2cos^22x +...

Use the comparison test to see if $$\sum_{1}^{\infty}{\left[n\left(n+1\right)\right]}^{-\frac{1}{2}} $$converges? I tried $$n+1 \gt n, \therefore n(n+1) \gt n^2 , \therefore {\left[n(n+1)\right]}^{\frac{1}{2}} \gt n, \therefore {\left[n(n+1)\right]}^{-\frac{1}{2}} \lt \frac{1}{n}$$ - no...

Homework Statement Homework Equations Series: R_{eq} = R_1 + ... + R_n Parallel: R_{eq} = \left(\frac{1}{R_1} + ... + \frac{1}{R_n} \right)^{-1} Charging Capacitor: I = I_0 e^{-t/RC} Charging Capacitor: \Delta V_C = \varepsilon (1- e^{-t/RC}) Charge: Q = C \Delta V_C The Attempt at a...

Homework Statement hello this question is discussed in 2009 but it is closed now If you invest £1000 on the first day of each year, and interest is paid at 5% on your balance at the end of each year, how much money do you have after 25 years? Homework Equations ## S_N=\sum_{n=0}^{N-1} Ar^n##...

Homework Statement In the arrangement shown,find the equivalent capacitance between A and B. Homework Equations Capacitance in parallel ##C##=##C_1##+##C_2##The Attempt at a Solution Supplied solution says As,we can clearly see that ,capacitors 10μF and 20μF are connected between same points...

Question: Why equations x(1-x)\frac{d^2y}{dx^2}+[\gamma-(\alpha+\beta+1)x]\frac{dy}{dx}-\alpha \beta y(x)=0 should be solved by choosing ##y(x)=\sum^{\infty}_{m=0}a_mx^{m+k}## and not ##y(x)=\sum^{\infty}_{m=0}a_mx^{m}##? How to know when we need to choose one of the forms. Also when I sum over...

Q. Show by induction that $ \sum_{1}^{\infty} \frac{1}{(2n-1)(2n+1)} = \frac{1}{2} $ So, start with base case n=1, $ S_1 = \frac{1}{(2-1)(2+1)} = \frac{1}{3}$? Maybe it's bedtime ...

Homework Statement All relevant data and variables are included in the image. The questions are also included in it. Homework Equations My questsion is just verification. I have attempted all the asked questions on the paper. Its frustrating as the papers don't include answers to check them...

1. Homework Statement Find the Fourier series of ##f(x) = \delta (x) - \delta (x - \frac{1}{2})## , ## - \frac{1}{4} < x < \frac{3}{4}## periodic outside. Homework Equations [/B] ##\int dx \delta (x) f(x) = f(0)## ##\int dx \delta (x - x_0) f(x) = f(x_0)##The Attempt at a Solution...

Homework Statement Approximate the integral to 3 decimal place accuracy via power series. ##\int_0^{1/2} x^2 e^{-x^2}\, dx ## Homework EquationsThe Attempt at a Solution ##x^2 e^{-x^2} = x^2 \sum_{n=0}^\infty \frac {(-x)^{2n}}{n!} = \sum_{n=0}^\infty \frac {x^{2n+2}}{n!}## ⇒ ##\int_0^{1/2}...

Homework Statement By considering the power series (good for |x| < 1) ##\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + x^4 +...## Describe how to manipulate this series in some way to obtain the result: ##\sum_{n=1}^\infty nx^n = \frac{x}{(1-x)^2}## Homework Equations Maclaurin...

My series foundation is really weak, and this spring I'll be taking Differential Equations. I know series plays a big role in solving some ODEs, so I'll be re-learning the material from Calc 2 to make sure I'm up to par for the class. I'll be working through Stewart's chapter on series but was...

Hello everyone, I'm working through some homework for a second year mathematical physics course. For the most part I am understanding everything however there is one step I do not understand regarding the steps taken to solve for the coefficients of the Legendre series. Starting with setting...

Hi there! I need a bit of help on a homework problem. The problem is about a voltage (V) across a circuit with a resistor (R) and and inductor (L). The current at time "t" is: I= (V/R)(1/e^(-RT/L) And the problem asks me to use Taylor series to deduce that I is approximately equal to (Vt/L) if...

Hey all! I am in Calculus 2, and we are starting to get into series. This may seem like an odd question, but on quizzes I seem to have difficulty identifying the type of series in order to be able to properly work it, and I'd like to have this down before I get to the test. Does anybody have a...

Homework Statement For which number x does the following series converge: http://puu.sh/lp50I/3de017ea9f.png Homework Equations abs(r) is less than 1 then it is convergent. r is what's inside the brackets to the power of n The Attempt at a Solution I did the question by using the stuff in...

Hello, Im currently in a Calc II class with unfortunately a bad professor (score of 2 on RateMyProfessor), so I often have to resort to outside sources to learn. Our class is currently on Sequences and Series which has been fine up until we hit the topic of relating Power Series and Functions...

Homework Statement Given the differential equation (\sin x)y'' + xy' + (x - \frac{1}{2})y = 0 a) Determine all the regular singular points of the equation b) Determine the indicial equation corresponding to each regular point c) Determine the form of the two linearly independent solutions...

I am in calc2 now and I just can't get excited about this series stuff. We went over many methods of testing for convergence/divergence and finally moved on to polar coordinates. Is series important in any type of CS field besides, I would imagine, creating software to solve series problems?

Homework Statement Using the CTFS table of transforms and the CTFS properties, find the CTFS harmonic function of the signal 2*cos(100*pi(t - 0.005)) T = 1/50 Homework Equations To = fundamental period T = mTo cos(2*pi*k/To) ----F.S./mTo---- (1/2)(delta[k-m] + delta[k+m]) The Attempt at...

Hello, I have been reviewing my textbook lately, and I came across a rather paradoxical statements. all of the convergence tests in my book state that the terms of the series has to be positive. However, when I solved this power series ∑(-1)n-1(xn/n), I found that it converges for -1<x≤1, but...

1. Homework Statement Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges Homework Equations 3/(1*2*3) + 3,/(2*3*4) + 3/(3*4*5) +...+ 3/n(n+1)(n+2) The Attempt at a Solution the first try, i tried using partial fraction which equals...

I can not find a solid explanation on this anywhere, so forgive me if this has been addressed already. Given something like y''+y'-(x^2)y=1 or y''+2xy'-y=x, how do I approach solving a differential with a power series solution when the differential does not equal zero? Would I solve the left...

So, in a section on applying Eigenvectors to Differential Equations (what a jump in the learning curve), I've encountered e^{At} \vec{u}(0) = \vec{u}(t) as a solution to certain differential equations, if we are considering the trial substitution y = e^{\lambda t} and solving for constant...

What is the relationship between the Fourier transform of a periodic function and the coefficients of its Fourier series? I was thinking Fourier series a special version of Fourier transform, as in it can only be used for periodic function and only produces discrete waves. By this logic, aren't...

I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series ... ) In Chapter 2: Linear Algebras and Artinian Rings, on Page 61 we find a definition of a refinement of a chain and a definition of a composition series. The relevant text on page 61 is as...

I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 7.1 Chain Conditions (for modules) ... I need some help in order to gain a full understanding of some remarks made in AMA on page 526 on modules in the context of chain conditions and...

I have made two posts recently concerning the composition series of groups and have received considerable help from Euge and Deveno regarding this topic ... in particular, Euge and Deveno have pointed out the role of the Correspondence Theorem for Groups (Lattice Isomorphism Theorem for Groups)...