In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor, who introduced them in 1715.
If zero is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).
Homework Statement
Homework Equations
All Fourier series trigonometric equations. I think we are required to use sigma function, integrals, etc.[/B]The Attempt at a Solution
We are currently working through our Fourier series revision studying integrals of periodic functions within K.A...
Q: Suppose ##u(x,t)## satisfies the heat equation for ##0<x<a## with the usual initial condition ##u(x,0)=f(x)##, and the temperature given to be a non-zero constant C on the surfaces ##x=0## and ##x=a##.
We have BCs ##u(0,t) = u(a,t) = C.## Our standard method for finding u doesn't work here...
I was given a function that is periodic about 2π and I need to plot it. I was wondering if there is a way to input a value and have mathematica generate a new graph with the number of iterations. The function is:
$$\sum_{n=1}^{N}\frac{sin(nx)}{n}$$ where n is an odd integer. I guess a better...
Hey, I have a quick question here from my assignment. I thought it did it right the first time but I got the wrong answer, and I can't possibly seem to find anything wrong with my solution. Any ideas?
Link to Question:
Imgur: The most awesome images on the Internet
So, I started by solving...
Homework Statement
Fourier series expansion of a signal f(t) is given as
f(t) = summation (n = -inf to n = +inf) [3/(4+(3n pi)2) ) * e j pi n t
A term in expansion is A0cos(6 pi )
find the value of A0
Repeat above question for A0 sin (6 pi t)
Homework Equations
Fourier expansion is summation...
Homework Statement
For example
cosh(x) = 1+x2/2!+x4/4!+x6/6!+...
Homework EquationsThe Attempt at a Solution
So plugging in x=0 you get that coshx = 1 at the origin. The approximate graph for the coshx function up to the second order looks like a 1+x2/2! graph, but what about graphing coshx...
Homework Statement
The problem from the textbook is:
Is it possible to connect resistors together in a way that cannot be reduced to some combination of series and parallel combinations?
Homework Equations
V = IR
kirchhoff's current law
kirchhoff's loop law
The Attempt at a Solution
I am...
Say you have two separate resistors in a circuit. When you close the switch on the circuit is it the electric field that flows through the circuit that effectively sets the voltage drops so a bigger voltage drop occurs across the higher resistor in proportion to its resistance such that the...
Homework Statement
Use a comparison test to determine whether this series converges:
\sum_{x=1}^{\infty }\sin ^2(\frac{1}{x}) Homework EquationsThe Attempt at a Solution
At small values of x:
\sin x\approx x
a_{x}=\sin \frac{1}{x}
b_{x}=\frac{1}{x}
\lim...
Homework Statement
\sum_{x=2}^{\infty } \frac{1}{(lnx)^9}
Homework EquationsThe Attempt at a Solution
x \geqslant 2
0 \leqslant lnx < x
0 < \frac{1}{x} < \frac{1}{lnx}
From this we know that 1 / lnx diverges and I wanted to use this fact to show that 1 / [(lnx) ^ 9] diverges but at k...
Homework Statement
Find the Fourier series of the function
f(x) =√(x2) -pi/2<x<pi/2 , with period pi
Homework EquationsThe Attempt at a Solution
I have tried attempting the question, but couldn't get the answer. uploaded my...
Homework Statement
Find the power series in x-x0 for the general solution of y"-y=0; x0=3.
Homework Equations
None.
The Attempt at a Solution
Let me post my whole work:
Assuming the resistance of a wire in a series circuit, consisting only of 1 component (e.g. filament lamp) and a battery, is negligible; does each Coulomb of charge commit all of its electrical potential energy, supplied by the battery's potential difference, as work done across the component...
Homework Statement
No actual problem, thinking about the telescoping series theorem and Grandi's series
For reference Grandi's series S = 1 - 1 + 1 - 1...
Homework Equations
[/B]
The telescoping series theorem in my book states that a telescoping series of the form (b1 - b2) + ... + (bn -...
Self Study
1. Homework Statement
Consider a periodic function f (x), with periodicity 2π,
Homework Equations
##A_{0} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)dx##
##A_{n} = \frac{2}{L}\int_{X_{o}}^{X_{o}+L}f(x)cos\frac{2\pi rx}{L}dx##
##B_{n} =...
I'm studying QFT in the path integral formalism, and got stuck in deriving the Schwinger Dyson equation for a real free scalar field,
L=½(∂φ)^2 - m^2 φ^2
in the equation,
S[φ]=∫ d4x L[φ]
∫ Dφ e^{i S[φ]} φ(x1) φ(x2) = ∫ Dφ e^{i S[φ']} φ'(x1) φ'(x2)
Particularly, it is in the Taylor series...
Homework Statement
Using the taylor series at point ##(x=0)## also known as the meclaurin series find the limit of the expression:
$$L=\lim_{x \rightarrow 0} \frac{1}{x}\left(\frac{1}{x}-\frac{cosx}{sinx}\right)$$
Homework Equations
3. The Attempt at a Solution [/B]
##L=\lim_{x \rightarrow 0}...
Homework Statement
Hello,
I am currently doing some holiday pre-study for signals analysis coming up next semester. I'm mainly using MIT OCW 6.003 from 2011 with some other web resources (youtube, etc).
The initial stuff is heavy on the old infinite series stuff, that seems often skimmed...
I have in my lecture notes that ##E_{k}(t=0) =1 ##,
##E_k (t)## the Eisenstein series given by:
##E_k (t) = 1 - \frac{2k}{B_k} \sum\limits^{\infty}_1 \sigma_{k-1}(n) q^{n} ##
##B_k## Bermouli number
##q^n = e^{ 2 \pi i n t} ##
context modular forms. Also have set ##lim t \to i\infty = 0## ...
So I am trying to figure out how much Wh I would have with these solar cells I have. Each solar cell is rated to have 2.8w. Does this mean if I have 40 of them I would have 112 wh? I am going to be putting the solar cells in series. Does this affect the power? I know adding in series is good...
Homework Statement
Find the values of the constant a for which the problem y''(t)+ay(t)=y(t+π), t∈ℝ, has a solution with period 2π
which is not identically zero. Also determine all such solutions
Homework Equations
With help of Fourier series I know that :
Cn(y''(t))= -n2*Cn(y(t))
Cn(y(t+π)) =...
Hi
I am supposed to find solution of $$xy''+y'+xy=0$$
but i am left with reversing this equation.
i am studying solution of a differential equation by series now and I cannot reverse a series in the form of:
$$ J(x)=1-\frac{1}{x^2} +\frac{3x^4}{32} - \frac{5x^6}{576} ...$$
$$...
[Please excuse the screengrabs of the fomulae - I'll get around to learning TeX someday!]
1. Homework Statement
Find the sum of this series (answer included - not the one I'm getting)
The Attempt at a Solution
So I'm trying to sum this series as a telescoping sum. I decomposed the fraction...
[mentor note: thread moved from non-hw forum to here hence no homework template]
Can someone explain to me how it is that
$$\sum_{n=a}^b (2n+1)=(b+1)^2-a^2$$
I thought it would be $$\sum_{n=a}^b (2n+1)=(2a+1)+(2b+1)$$
but I am clearly very wrong. I would greatly appreciate any help.
Homework Statement
A fishery manager knows that her fish population naturally increases at a rate of 1.4% per month, while 119
fish are harvested each month. Let Fn be the fish population after the nth month, where F0 = 4500 fish. Assume that that process continues indefinitely. Use the...
Homework Statement
I have been working through “Basic Concepts in Relativity and Early Quantum Theory” by Resnick and Halliday. I've read about and done most of the problems about time-dilation, length-contraction, and Doppler effect. But then I got to problem 2-76, and I’ve been swirling in...
<OP warned about not using the homework template>
Obtain a series solution of the differential equation x(x − 1)y" + [5x − 1]y' + 4y = 0Do I start by solving it normally then getting a series for the solution or assume y=power series differentiate then add up the series?
I did the latter and...
< Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown >
Hi all. I'm completely new to these forums so sorry if I'm doing anything wrong.
Anyway, I have this question...
Find the Fourier series for the periodic function
f(x) = x^2 (-pi < x < pi)...
Hi, I'm starting to studying Fourier series and I have troubles with one exercises of complex Fourier series with
f(t) = t:
$$t=\sum_{n=-\infty }^{\infty } \frac{e^{itn}}{2\pi }\int_{-\pi}^{\pi}t\: e^{-itn} dt$$
$$t=\sum_{n=-\infty }^{\infty } \frac{cos(tn)+i\, sin(tn)}{2\pi...
Hi.
I helped my neighbour putting up (quite old, no LEDs) strings of Christmas lights and noted that some of them (different brands) are connected in parallel, others in series.
Inevitably, we found several of the serial strings not working due to defective bulbs. We replaced the visibly...
Suppose that the Taylor series of a function ##f: (a,b) \subset \mathbb{R} \to \mathbb{R}## (with ##f \in C^{\infty}##), centered in a point ##x_0 \in (a,b)## converges to ##f(x)## ##\forall x \in (x_0-r, x_0+r)## with ##r >0##. That is
$$f(x)=\sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n...
$\tiny{206.r2.11}$
$\textsf{find the power series represntation for
$\displaystyle f(x)=\frac{x^7}{3+5x^2}$
(state the interval of convergence),
then find the derivative of the series}$
\begin{align}
f(x)&=\frac{x^7}{3}\implies\frac{1}{1-\left(-\frac{5}{3}x^2\right)}&(1)\\...
We made a RLC circuit in the lab and took some values of R and LC Voltage while we changed the frequency.
So the experimental data seem to suggest that at resonance (VLC=18 mV : min) the Voltage of the resistor is 834 mV. But the initial voltage given was measured 1.426 V (All Values rms)...
Homework Statement
and the solution (just to check my work)
Homework Equations
None specifically. There seems to be many ways to solve these problems, but the one used in class seemed to be partial fractions and Taylor series.
The Attempt at a Solution
The first step seems to be expanding...
I have this:
$$ \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. ?
$\textsf{a. Find the first four nozero terms of the Maciaurin series for the given function} \\$
\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...
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...