Series Definition and 998 Threads

If we can use Regression analysis to forecast, why do we use “Time Series Decomposition”? What's the difference between the 2? Thanks

I have this: $$ \sum_{n = 1}^{\infty} \frac{n^n}{3^{1 + 3n}}$$ And I need to determine if it is convergent or divergent. I try the limit comparison test against: $$ \frac{1}{3^{1 + 3n}}$$. So I need to determine $$ \lim_{{n}\to{\infty}} \frac{3^{1 + 3n} \cdot n^n}{3^{1 + 3n}}$$ Or $$...

$\textsf{a. Find the first four nozero terms of the Maciaurin series for the given function} \\$ \begin{align} f^0(x)&=\ln{ (6 x + 1)} &\therefore f^0(a)&=0\\ f^1(x)&=\frac{6}{(6 x + 1)} &\therefore f^1(a)&=6\\ f^2(x)&= \frac{-36}{(6 x + 1)^2} &\therefore f^2(a)&=-36\\ f^3(x)&= \frac{432}{(6 x...

I have $$\sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$$ I'm trying the limit comparison test, so I let $$ b = \frac{1}{n^{\frac{9}{8}}}$$ and $a = \sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$ $\frac{a}{b} = (lnn)^ {12}$ therefore I know the limit of this as n...

Homework Statement consider ##f## a meromorphic function with a finite pole at ##z=a## of order ##m##. Thus ##f(z)## has a laurent expansion: ##f(z)=\sum\limits_{n=-m}^{\infty} a_{n} (z-a)^{n} ## I want to show that ##f'(z)'/f(z)= \frac{m}{z-a} + holomorphic function ## And so where a...

$\textsf{a. Find the first three nonzero terms of the Taylor series $a=\frac{3\pi}{4}$}$ \begin{align} \displaystyle f^0(x)&=\sin{x} &\therefore \ \ f^0(a)&=\sin{x} \\ f^1(x)&=\cos{x} &\therefore \ \ f^1(a)&= -\frac{\sqrt{2}}{2}\\ f^2(x)&=- \sin{x}&\therefore \ \ f^2(a)&=\frac{\sqrt{2}}{2} \\...

multiple 220V to 5v, 2A ac to dc adapters connected to the same 2-phase input terminals input voltage is 220v domestic supply. is it safe to i join output in series to obtain 10v /15v/25v etc. ?

$\textsf{a. Find the first four nonzero terms of the Taylor series $a=1$}$ \begin{align} \displaystyle f^0(x)&=6^{x} &\therefore \ \ f^0(a)&= 6 \\ f^1(x)&=6^{x}\ln(6) &\therefore \ \ f^1(a)&= 6\ln(6) \\ f^2(x)&={6^{x}\ln(6)^2} &\therefore \ \ f^2(a)&= {12\ln(6)} \\ f^3(x)&={6^{x}\ln(6)^3}...

$\tiny{206.11.3.39}$ $\textsf{a. Find the first four nonzero terms of the Taylor series $a=0$}$ \begin{align} \displaystyle f^0(x)&=(1+x)^{-2} &\therefore \ \ f^0(a)&= 1 \\ f^1(x)&=\frac{-2}{(x+1)^3} &\therefore \ \ f^1(a)&= -2 \\ f^2(x)&=\frac{6}{(x+1)^4} &\therefore \ \ f^2(a)&= 6 \\...

$\textsf{a. Find the first four nozero terms of the Maciaurin series for the given function} \\$ \begin{align} a&=0 \\ f(x)&=(-5+x^2)^{-1} \\ \\ f^0(x)&=(-5+x^2)^{-1}\therefore f^0(a) = 1 \\ f^1(x)&=\frac{-2x}{(x^2-5)^2} \therefore f^1(a) = 0 \\ f^2(x)&=\frac{2(3x^3+5)}{(x^2-5)^3} \therefore...

Is the series of numbers 2,3,5,8,13,21 ... a fibronacci sequence ? Because it doesn't start with 1 , but it fulfills the explicit formula .

Homework Statement A metal wire of resistance R is cut into two pieces of equal length. The two pieces are connected together side by side. What is the resistance of the two connected wires? Homework Equations RSeries = R1 + R2 + ... RParallel = (1/R1 + 1/R2 + ...)-1 The Attempt at a Solution...

Homework Statement Using Taylor series, Find a polynomial p(x) of minimal degree that will approximate F(x) throughout the given interval with an error of magnitude less than 10-4 F(x) = ∫0x sin(t^2)dt Homework Equations Rn = f(n+1)(z)|x-a|(n+1)/(n+1)![/B] The Attempt at a Solution I am...

So, ∫x/(1-x)... can I solve this as a power series ∫(x*Σ x^n) = ∫(Σ x^(n+1))= (1/(n+2)*Σ x^(n+2))? Is this correct? I know there is other ways to do it... But should this be correct on a test? This solution is more fun..

Homework Statement Determine the Taylor series for the function below at x = 0 by computing P5(x) f(x) = cos(3x2) Homework Equations Maclaurin Series for degree 5 f(0) + f1(0)x + f2(0)x2/2! + f3(0)x3/3! + f4(0)x4/4! + f5(0)x5/5! The Attempt at a Solution I know how to do this but attempting...

hi, I try to calculate the integral $$\int_{0}^{1}log(\Gamma (x))dx$$ and the last step To solve the problem is: $$1 -\frac{\gamma }{2} + \lim_{n\rightarrow \infty } \frac{H_{n}}{2} + n + log(\Gamma (n+1)) - (n+1)(log(n+1))$$ and wolfram alpha tells me something about series expansion at...

Homework Statement [/B]Homework Equations an= bn - bn+1 which is already in the problem The Attempt at a Solution [/B] i did partial fractions but then i got stuck at 16/12 [4n-5] - 16/12 [4n+7] that part about bn confuses me please someone explain in detail

Hi, As part of showing that ##E^*_{2}(-1/t)=t^{2}E^*_{2}(t)## where ##E^*_{2}(t)= - \frac{3}{\pi I am (t) } + E_{2}(t) ## And since I have that ##t^{-2}E_{2}(-1/t)=E_{2}(t)+\frac{12}{2\pi i t} ## I conclude that I need to show that ##\frac{-1}{Im(-1/t)}+\frac{2t}{i} = \frac{-t^{2}}{Im(t)} ##...

$\tiny{s4.12.9.13}$ $\textsf{find a power series reprsentation and determine the radius of covergence.}$ $$\displaystyle f_{13}(x) =\frac{1}{(1+x)^2}=\frac{1}{1+2x+x^2}$$ $\textsf{using equation 1 }$ $$\frac{1}{1-x} =1+x+x^2+x^3+ \cdots =\sum_{n=0}^{\infty}x^n \, \, \left| x \right|<1$$...

Hi All, I've been reading up on hybrid vehicles and how the Honda Insight has an electric motor attached directly to the gas engine driveshaft. Their combined output causes the wheels to spin. My question is in regards to when the electric motor is off but the gas engine is on. Why doesn't the...

Homework Statement I want to show that ## \sum\limits_{n=1}^{\infty} log (1-q^n) = -\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty} \frac{q^{n.m}}{m} ##, where ##q^{n}=e^{2\pi i n t} ## , ##t## [1] a complex number in the upper plane.Homework Equations Only that ## e^{x} =...

It seems to me that convergence rounds away the possibility of there being a smallest constituent part of reality. For instance, adding 1/2 + 1/4 + 1/8 . . . etc. would never become 1, since there would always be an infinitely small fraction that made the second half unreachable relative to the...

has Fourier used sin(x) and cos(x) in his series because "there must be such interval [a,b] where integral of "some function"*sin(x) on that interval will be zero?" so based on that he concluded that any function can be represented by infinite sum of sin(x) and cos(x) cause they are "orthogonal"...

Homework Statement Find the sum of the given infinite series. $$S = {1\over 1\times 3} + {2\over 1\times 3\times 5}+{3\over 1\times 3\times 5\times 7} \cdots $$ 2. Homework Equations The Attempt at a Solution I try to reduce the denominator to closed form by converting it to a factorial...

Question ∞ ∑ tan(1/n) n = 1 Does the infinite series diverge or converge? Equations If limn → ∞ ≠ 0 then the series is divergent Attempt I tried using the limit test with sin(1/n)/cos(1/n) as n approaches infinity which I solved as sin(0)/cos(0) = 0/1 = 0 This does not rule out anything and I...

I have a set of data that I've been working with that seems to be defined by the sum of a set of exponential functions of the form (1-e^{\frac{-t}{\tau}}). I've come up with the following series which is the product of a decay function and an exponential with an increasing time constant. If this...

Hi folks, I was wondering if there are books that explain how to solve differential equations using infinite series. I know it is possible to do it since Poincaré used that method. Do you know which ones are the best? I find books on infinite series but they talk just about series...

Need help on Fourier series! Been stuck on this questions, it is too tough for me!

Homework Statement The problem asks to use a substitution y(x) to turn a series dependent on a real number x into a power series and then find the interval of convergence. \sum_{n=0}^\infty ( \sqrt{x^2+1})^n \frac{2^n }{3^n + n^3} Homework Equations After making a substitution, the book...

How exactly do you tell if two elements are in series or parallel? I know that if two elements share two of the same extraordinary nodes then they are in parallel: But in this example I do not see that. I know for certain that the answer for Rt is: [ ( (R1 || R2) + R3) || R4 ] + R5

Question 1: Find the Laurent series of \cos{\frac{1}{z}} at the singularity z = 0. The answer is often given as, \cos\frac{1}{z} = 1 - \frac{1}{2z^2} + \frac{1}{24z^4} - ... Which is the MacLaurin series for \cos{u} with u = \frac{1}{z}. The MacLaurin series is the Taylor series when u_0 = 0...

I was studying the derivation for taylor series in ℝ##^n## on my book and I have some trouble understanding a passage; it's the very beginning actually: ##f : A## ⊆ ℝ##^n## → ℝ ##f ## ∈ ##C^2(A)## ##x_0## ∈ ##A## "be ##g_{(t)} = f_{(x_0 + vt)}## where v is a generic versor, then we have...

$\tiny{242.WS10.a}$ \begin{align*} &\textsf{use the techniques of geometric series} \\ &-\textsf {telescoping series, p-series, n-th term } \\ &-\textsf{divergence test, integral test, comparison test,} \\ &-\textsf{limit comparison test,ratio test, root test, } \\ &-\textsf {absolute...

Does 0.999... equals 1? I know that this is a very basic well known concept but recently I stumbled across a video on Youtube in which the creator argues that the two are not equivalent I posted a comment arguing that in the case of Infinite sum of Σn=0 9(1/10)n you can find the sum of the...

The Chicago Cubs have just won the National League championship, and will play in the World Series for the first time since 1945. They last won the Series in 1908. http://www.cnn.com/2016/10/22/us/chicago-cubs-world-series-bid/index.html Normally, I would be delighted to cheer them on, but...

I have the following laplace function F(s) = (A/(s + C)) * (1/s - exp(-sα)/s)/(1 - exp(-sT)) I think that the inverse laplace will be- f(t) = ((A/C)*u(t) - (A/C)*exp(-Ct)*u(t)) - ((A/C)*u(t-α) - (A/C)*exp(-C(t-α))*u(t-α)) and f(t+T)=f(t) Now I want to find the Fourier series expansion of f(t)...

$\tiny{206.10.3.54}$ $\text{Write the repeating decimal first as a geometric series} \\$ $\text{and then as fraction (a ratio of two intergers)} \\$ $\text{Write the repeating decimal as a geometric series} $ $6.94\overline{32}=6.94323232 \\$ $\displaystyle A.\ \ \...

Homework Statement Link: http://i.imgur.com/klFmtTH.png Homework Equations a_0=\frac{1}{T_0}\int ^{T_0}_{0}x(t)dt a_n=\frac{2}{T_0}\int ^{\frac{T_0}{2}}_{\frac{-T_0}{2}}x(t)cos(n\omega t)dt \omega =2\pi f=\frac{2\pi}{T_0} The Attempt at a Solution Firstly, x(t) is an even function because...

$\displaystyle\text{if} \left| r \right|< 1 \text{ the geometric series } a+ar+ar^2+\cdots ar^{n-1}+\cdots \text{converges} $ $\displaystyle\text{to} \frac{a}{(1-r)}.$ $$\sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{(1-r)}, \ \ \left| r \right|< 1$$ $\text{if} \left| r \right|\ge 1 \text{, the series...

Would it be possible to couple and electric motor to gas engine in series so both would increase the overall power.I currently building a twin engine car with two gas engines coupled together crank to crank like the old school twin dragsters.I would like to connect a large D.C. Motor in front of...

So all these college classes are really a growing experience for my teenage boys (home schooled). Last night my older son kept us up late persevering on a Calculus problem. Now, I remember a lot about sequences and series from my own days in Calculus and from teaching Calc 1, 2, and 3 at the Air...

Hello. Homework Statement What is the impedance of a circuit in which resistor and inductor are connected in series with each other, and a capacitor is in parallel with them? How should I sum the voltages in order to find the impedance? 2. Homework Equations V = IZ IZ = VR + VC + VL ? The...

Consider the Taylor series expansion of ##e^{-x}## as follows: ##\displaystyle{e^{-x}=1-x+\frac{x^{2}}{2}-\frac{x^{3}}{6}+\dots}## For ##x>0##, the partial sums ##1##, ##1-x##, ##\displaystyle{1-x+\frac{x^{2}}{2}}## bound ##e^{-x}## from above and from below alternately. How do I prove this?

Homework Statement The emission wavelengths of hydrogen-like atoms are related to nuclear charge. How do they scale as a function of Z? What are the longest and shortest wavelengths in the Balmer series for ##Li^{2+}##? Homework Equations ##E_n = -\frac{R}{n^2}## (1) ##a_0 =...

$\tiny{242.ws8.d}$ $$\displaystyle L_d=\lim_{x \to \infty} \left[\frac{\arctan{(n)}}{\pi +\arctan{(n)}}\right] =\frac{1}{3}$$ $\text{L' didn't work}$ ☕

Homework Statement To Determine Whether the series seen below is convergent or divergent. Homework Equations ∑(n/((n+1)(n+2))) From n=1 to infinity. The Attempt at a Solution Tried to use the comparison test as the bottom is n^2 + 3n + 2, comparing to 1/n. However, this does not work as the...

Hi, I am trying to find out series and shunt resistance of solar cell from I-V curve which starts from Isc and going to Voc. I read some papers and for single illumination mostly said Rs= -(dv/dI)V=Voc and Rsh= -(dv/dI)I=Isc. so I am finidng Rs from slope between points from Vmax to Voc, and...

I have always been a bit confused about how changing indices in a summations changes the resulting closed formula. Take this geometric series as an example: ##\displaystyle \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^n}##. Putting it into summation notation, we have...

Homework Statement Find the impedance (Z) of the circuit in the shown figure (R and L in series, and C in parallel with them). A circuit is said to be in resonance if Z is real, find ω in terms of R, L, and C? Homework Equations VR= RIeiωt VL= LiωIeiωt VC= (Ieiωt)/(iωc) What I mean by I is I...

Homework Statement Expand the function f(z)=1/z(z-2) in a Laurent series valid for the annual region 0<|z-3|<1 Homework Equations I know 1/z(z+1) = 0.5(1/(z-2)) - 0.5(1/z) Taylor for 0.5(1/(z-2)) is : ∑(((-1)k/2) * (z-3)k) (k is from 0 to ∞)For the second 0.5(1/z) the answer is a...