Series Definition and 998 Threads

Homework Statement I am going over problems for exam study - here is the question with my submitted solution. Anything helps, just trying to correct mistakes so I can study the problems. Two capacitors C=3mF, C=2mF are initially discharged. They are connected in series and then the two ends...

Homework Statement [/B] There are three problems that I am struggling with. 1. ∑[k2(x-2)k]/[3k] 2. ∑[(x-4)n]/[(n)(-9)n] 3. ∑[2k(x-3)k]/[k(k+1)] The Attempt at a Solution On the first two I am having problems finding the end-points of the interval of convergence. I use the ratio test. 1...

Homework Statement Use a laurent series to find the indicated residue f(z)=e^{\frac{-2}{z^2}} Homework EquationsThe Attempt at a Solution So I expand the series as follows 1-\frac{2}{z^2}+\frac{2}{z^4} ... my book says the residue is 0 , is this because there is no residue term ? the...

Homework Statement "Find the recurrence relation in the power series solution for ##y''-xy'-y=0## centered about ##x_0=1##." Homework Equations ##y=\sum_{n=0}^\infty a_nx^n## Answer as given in book: ##(n+2)a_{n+2}-a_{n+1}-a_n=0## The Attempt at a Solution ##y=\sum_{n=0}^\infty a_n(x-1)^n##...

Hi Two lightbulbs in series, one with 50W one with 100W which is brighter. I have two different solutions and can't see my error. Using PR=V^2 and I^2=P/V 50/√50R1=100/√100R2 with the same current and R2=2R1. Using PR=V^2 and the same voltage across both bulbs yields 50R1=100R2 or R1=2R2. Which...

Homework Statement Determine the order of the poles for the given function. f(z)=\frac{1}{1+e^z} Homework EquationsThe Attempt at a Solution I know if you set the denominator equal to zero you get z=ln(-1) But if you expand the function as a geometric series , 1-e^{z}+e^{2z}... I...

Homework Statement WE have a thermally insulated metallic bar (from enviroment/surroundings) . It has a temperature of 0 ºC. At t=0 two thermal sources are applied at either end, the first being -10 ºC and the second being 10 ºC. Find the equation for the temperature along the bar T(x,t), in...

Homework Statement Show that ##\displaystyle \frac{1}{1+x^2} = \frac{1}{x^2} - \frac{1}{x^4} + \frac{1}{x^6} - \frac{1}{x^8} + \cdots## Homework EquationsThe Attempt at a Solution I know that the power series expansion of ##\displaystyle \frac{1}{1+x^2}## about ##x=0## is ##1-x^2 + x^4 - x^6 +...

Homework Statement expand f(z)=\frac{1}{z(z-1)} in a laurent series valid for the given annular domain. |z|> 3 Homework EquationsThe Attempt at a Solution first I do partial fractions to get \frac{-1}{3z} +\frac{1}{3(z-3)} then in the second fraction I factor out a z in the denominator...

Take the following diagram of 4 1.5V batteries connected in series to creat and net voltage of 6V (the numbers are of no significance here). If we were to short circuit the system by connecting battery 1 to battery 4 (or run the current through a load), wouldn't there be electrons traveling...

Homework Statement series from n = 1 to infinity, (ne^(-n)) Homework EquationsThe Attempt at a Solution I want to use integral test. I know this function is: positive (on interval 1 to infinity) continous and finding derivative of f(x) = xe^(-x) I found it to be ultimately decreasing. So...

$\tiny{242.tr.05}$ Use the integral test to determine if a series converges. $\displaystyle \sum_{n=1}^{\infty}\frac{1}{\sqrt{e^{2n}-1}}$ so... $\displaystyle \int_{1}^{\infty} \frac{1}{\sqrt{e^{2n}-1}}\, dn =\int_{1}^{\infty} (e^{2n}-1)^{1/2} \, dn $ so $u=e^{2n}-1\therefore du=2e^{2n}$

Homework Statement I've begun going through Boas' Math Methods in the Physical Sciences and am stuck on problem 1.15.25. The problem is to evaluate ## \lim_{x\to \infty } x^n e^{-x} ## By using the Maclaurin expansion for ##e^{x}##. Homework Equations We know the Maclaurin expansion for the...

Homework Statement I want to find the Fourier series of the sawtooth function in terms of real sine and cosine functions by using the formula: $$f_p (t)=\sum^\infty_{k=-\infty} c_k \exp \left(j2\pi \frac{k}{T}t \right) \tag{1}$$ This gives the Fourier series of a periodic function, with the...

Homework Statement Evaluate the indefinite integral as a power series. What is the radius of convergence (R)? ##\int x^2ln(1+x) \, dx## Book's answer: ##\int x^2ln(1+x) dx = C + \sum_{n=1}^\infty (-1)^n \frac {x^{n+3}} {n(n+3)}; R = 1## Homework Equations Geometric series ##\frac {1} {1-x} =...

Homework Statement It is the driven series RLC circuit. It is given in the following images. It is from the section 12.3 in this note. Homework Equations The differential equation, as given by 12.3.3, is ##L \frac{d^2 Q}{d t^2} + R \frac{d Q}{d t} + \frac{Q}{C} = V_0 \sin{(\omega t)}##...

Homework Statement Hi I am trying to understand this http://math.stackexchange.com/questions/341406/how-do-i-obtain-the-laurent-series-for-fz-frac-1-cosz4-1-about-0 So the long division yields...

Need help with a homework question! The question gives: The first three terms of a geometric sequence are sin(x), sin(2x) and 4sin(x)cos^2(x) for -π/2 < x < π/2. First I had to find the common ratio which is 2cos(x) Then the question asks to find the values of x for which the geometric series...

I'm trying to determine if \sum_{n=1}^{\infty}\frac{{n}^{10}}{{2}^{n}} converges or diverges. I did the ratio test but I'm left with determining \lim_{{n}\to{\infty}}\frac{(n+1)^{10}}{2n^{10}} Any suggestions??

Homework Statement Given n>=0, create an array length n*n with the following pattern, shown here for n=3 : {0, 0, 1, 0, 2, 1, 3, 2, 1} (spaces added to show the 3 groups). Homework EquationsThe Attempt at a Solution public int[] squareUp(int n) { int length = n*n; int[] completeArry...

Hi, I stamped at a series expansion. It is probably Taylor. Would you explain it? It's in the vid. https://confluence.cornell.edu/display/SIMULATION/Big+Ideas%3A+Fluid+Dynamics+-+Differential+Form+of+Mass+Conservation I understand equation 1 in the picture but I do not understand 2. I...

Today a professor of mine who teaches Electrical machines told us that a DC series motor should not be started without load. I wonder why is that so. Please provide a detailed explanation of this PF members. Thank you very much in advance.

<Moderator's note: moved from a technical forum, so homework template missing> Hi. I have solved the others but I am really struggling on 22c. I need it to converge for |z|>2. This is the part I am really struggling with. I am trying to get both fractions into a geometric series with...

This is my first time posting so forgive me if I have it in the wrong place, i'm trying to find a solution to the following that I can stick into either excel or a VBA script. It has been 25 years since I looked at any serious maths and I'm stumped. I can find and digest e^-(n^2y) but can't...

I found this series, when I tried to evaluate the net Newtonian gravitational force on a mass at rest upon one vertex of a cube while all the other masses were arranged on an orthogonal lattice inside the cube: ## \sum\limits_{k=1}^{\infty} \sum\limits_{j=0}^{\infty} \sum\limits_{i=0}^{\infty}...

Hello, can we make a Fourier series expansion of a (increasing or decreasing) step function ? like the one that I attached here. I just want to know the idea of that if it is possible.

Homework Statement Let ## f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx) ## What can be said about the coefficients ##a_n## and ##b_n## in the following cases? a) f(x) = f(-x) b) f(x) = - f(-x) c) f(x) = f(π/2+x) d) f(x) = f(π/2-x) e) f(x) = f(2x) f) f(x) = f(-x) =...

In the power series below, I've used the ratio test and at the end I got |x-2| times infinity which is >1 so it diverges.. and in this case there is no interval of convergence because it's times inifnity.. How did he conclude that it converges at x=2??

I've 2 questions 1) Why do we take absolute of the power series? 2) I don't get why the interval of convergence is from -inifinity to +infinity. You can find the problem below.

Homework Statement Design a series RLC filter for 10kHz using an 0.01mF capacitor. Homework Equations / 3. The Attempt at a Solution how would the circuit actually look if drawn out here? [/B]

Can I use the divergence test on the partial sum of the telescoping series? Lim n>infinity an if not equal zero then it diverges The example below shows a telescoping series then I found the partial sum and took the limit of it. My question is shouldn't the solution be divergent? Since the...

Homework Statement and in this case we have, [PLAIN]http://tutorial.math.lamar.edu/Classes/CalcII/ConvergenceOfSeries_files/eq0016MP.gif[PLAIN]http://tutorial.math.lamar.edu/Classes/CalcII/ConvergenceOfSeries_files/empty.gif Homework Equations I can not see how they get either of...

Homework Statement I know that ∑n=1 to infinity (sin(p/n)) diverges due using comparison test with pi/n, despite it approaching 0 as n approaches infinity. However, an alternating series with (-1)^n*sin(pi/n) converges. Which does not make sense because it consists of two diverging functions...

Hi, I would like to as you you help please with finding whether the following three series converge. \sum_{1}^{\infty} (-1)kk3(5+k)-2k $$\sum_{k=1}^\infty(-1)^kk^3(5+k)^{-2k}$$ \sum_{2}^{\infty} sin(Pi/2+kPi)/(k0.5lnk) $$\sum_{k=2}^\infty\frac{\sin\left(\frac{\pi}{2}+k\pi\right)}{\sqrt k\ln...

Homework Statement \begin{equation} (1-x)y^{"}+y = 0 \end{equation} I am here but do not understand how to combine the two summations: Mod note: Fixed LaTeX in following equation. $$(1-x)\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n+\sum_{n=0}^{\infty}a_nx^n = 0$$

Hi all, I have a trigonometric function series $$f(x)={1 \over 2}{\Lambda _0} + \sum\limits_{l = 1}^\infty {{\Lambda _l}\cos \left( {lx} \right)} $$ with the normalization condition $$\Lambda_0 + 2\sum\limits_{l = 1}^\infty {{\Lambda _l} = 1} $$ and ##\Lambda_l## being monotonic decrescent...

$\tiny{10.6.44}\\$ $\textsf{Does $S_n$ Determine whether the series converges absolutely, conditionally or diverges.?}\\$ \begin{align*}\displaystyle S_n&= \sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}+\sqrt{n+6}}\\ \end{align*} $\textit {apparently the ratio and root tests fail}$

Homework Statement f(x) = -1, -π ≤ x ≤ 0 2, 0 ≤ x ≤ π Given this find the Fourier series using both $$a) \sum_{n=-∞}^\infty a_n e^{inx}$$ $$b) \sum_{n=0}^\infty [A_n cos(nx) + B_n sin(nx)]$$ Homework Equations $$a_o = \frac {1} {2L} \int_{-L}^L f(t) \, dt $$ $$a_n = \frac {1} {L}...

{\displaystyle \sum_{n=1}^{\infty}a_{n}} is converage, For N\in \mathbb{N}\sum_{n=N+1}^{\infty}an is also converage proof that \lim_{N\rightarrow\infty}(\sum_{n=N+1}^{\infty}an)=0 {\displaystyle \sum_{n=1}^{\infty}a_{n}} is converage, For N\in \mathbb{N} \sum_{n=N+1}^{\infty}an is...

$\tiny{10.5.55}$ $\textsf{ Does the following series converge or diverge?}$ \begin{align*}\displaystyle S_{n}&=\sum_{n=1}^{\infty}\frac{10^n n!n!}{(2n)!} \\ &= \end{align*} $\textit{ratio test?}$ :cool:

$\textsf{Find the sum of the series}\\$ \begin{align*}\displaystyle S_{n}&=\sum_{n=1}^{\infty} \frac{4}{(4n-1)(4n+3)}=\color{red}{\frac{1}{3}} \\ \end{align*} $\textsf{expand rational expression } $ \begin{align*}\displaystyle \frac{4}{(4n-1)(4n+3)}...

Homework Statement Derive the expression for coefficients of Fourier series in exponential form for the sequence of rectangular pulses (with amplitude A, period T and duration θ) shown in this image: Derive the expression for signal power depending on the coefficients of Fourier series...

Homework Statement What is the value of ## \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + ... ## ? Homework Equations [/B] I have no idea since it's neither a geometric nor arithmatic seriesThe Attempt at a Solution [/B] My Calculus purcell book tells me that it is e - 1 ≈...

Poster warned that the homework template is not optional. Determine if they are convergent or divergent, If it converges find the sum: ∞ ∑ 3^(n-1) 2^n n=1 ∞ ∑ ln(1/n) n=1 ∞ ∑ tan^n ( π/6) n=1 I tried to find information on how to solve them but I couldn't, thanks for the help

Homework Statement I have a couple of series where I need to find out if they are convergent (absolute/conditional) or divergent. Σ(n3/3n Σk(2/3)k Σ√n/1+n2 Σ(-1)n+1*n/n^2+9 Homework Equations Comparison Test Ratio Test Alternating Series Test Divergence Test, etc The Attempt at a...

In this alternate universe, Earth is the same as back home--8,000 miles wide, 25,000 around, six sextillion tons, orbiting a G-type main-sequence star from a distance of 93 million miles. But here, the similarities end. MOON DIAMETER--3,273 miles MASS--0.025x that of Earth DISTANCE FROM...

Hi, I am trying to solve this series generally: the series: 3 7 12 18 25. i tried using x(n) = 3 + 4n. But this doesn't work.. Please help.

Has anyone read the 7-book series http://amzn.to/2lwgn66 by Andrew Thomas? Just wondering what you think of his conjectures / speculations at the final sections of each book, i.e. on the link between relativity and quantum mechanics, equation of the universe, etc... I like that he...

Homework Statement There are three capacitors C1 = 2 uF, C2 = 4 uF, C3 = 6 uF. Each of these capacitors were connected to 200-V voltage source so every capacitor has been fully charged. Then, the three capacitors are connected like the image above. When S1 and S2 are closed, but S3 is...

Homework Statement i have this function \begin{equation} f(t) = e^t \end{equation} Homework Equations [/B] the Fourier seria have the form \begin{equation} f(t) = \sum C_{n} e^{int} \end{equation}The Attempt at a Solution } [/B] so i need to find the coeficients $c_{n}$ given by...