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).
I understand algebraically that when capacitors are in series, the total capacitance is less than any individual capacitance, but I do not understand this intuitively. How can this be possible? Shouldn't more capacitors equal more capacitance?
Homework Statement
In the following problem I am trying to extend the function $$f(x) = x $$ defined on the interval $$(0,\pi)$$ into the interval $$(-\pi,0)$$ as a even function. Then I need to find the Fourier series of this function.Homework EquationsThe Attempt at a Solution
So I believe I...
Hello.
I have this function ## v(x) = -\sum_{i=1} x^i \sqrt{2}^{i-2} \int_{-\infty}^{\infty} m^{i-1} \cosh(m)^{-4} dm## which I can not seem to figure out how to simplify.I tried looking at some partial integration but repeated integration of ## \cosh ## gives polylogarithms which seemed to...
In the following question I need to find the Fourier cosine series of the triangular wave formed by extending the function f(x) as a periodic function of period 2
$$f(x) = \begin{cases}
1+x,& -1\leq x \leq 0\\
1-x, & 0\leq x \leq 1\\\end{cases}$$
I just have a few questions then I will be able...
Homework Statement
With just about any problem asking for "rate at which source is delivering electrical energy to the circuit" or "find the power of the circuit" in a LRC circuit, I get that you have to calculate for the average power. But the multiple equations confuse me - sometimes in...
Homework Statement
Find trigonometric Fourier series for ##f(x)=|x|##, ##x∈[−\pi, \pi]##, state points where ##F(x)## fail to converge to ##f(x)##.
Homework Equations
##F(x) = \frac{a_0}{2}+\sum\limits_{n=1}^\infty a_ncos(\frac{n\pi x}{L})+b_nsin(\frac{n\pi x}{L})##...
Homework Statement
Consider the Fourier series of a signal given by
$$x(t)=\sum_{k=-\infty}^{\infty} a_ke^{jk\omega_0t}$$
Let's consider an approaches to this series given by the truncated series.
$$x_N(t)=\sum_{k=-N}^{N} a_ke^{jk\omega_0t}$$
a- Show that if $x(t)$ is real then the series...
Let h(h(x)) = exp(x), where h(⋅) is holomorphic in the whole ℂ plane.
I want an extension of the domain of exp(⋅) and of h(⋅) so that
we can find values of these functions for x = Aleph(0).
Homework Statement I am having a slight issue with generating function of legendre polynomials and shifting the sum of the genertaing function.
So here is an example:
I need to derive the recurence relation ##lP_l(x)=(2l-1)xP_{l-1}(x)-(l-1)P_{l-2}##
so I start with the following equation...
I'm trying to use Maxima to examine the error in a Fourier series as the number of terms increases. I've figured out how to produce a Fourier series and plot partial sums, but this has me stumped.
If anyone experienced with the Maxima CAS has some insight into this, I would greatly appreciate...
Homework Statement
Find Fourier coefficients of the periodic function whose template is x(t) where the Fourier Transform of x(t) is X(f) = (1-f^2)^2 where \left|f\right|<1 and period T_0= 4.
Homework Equations
FC=\hat x_T(k,T_0)=\sum_{k=-\infty}^\infty\frac{1}{T_0}X\left(k/T_0\right)
The...
Homework Statement
Hi! A battery of emf 12 V and negligible internal resistance is connected to a resistor of constant resistance 6 Ω, an ideal ammeter and an ideal voltmeter. The voltmeter and ammeter are in series with the cell and the resistor. What is the reading on each?
Homework...
i have attached the problem set.
I have done the first three problems but number 4 is very difficult.
Can someone help me out?
Thanks
[Editor's note: The PDF below contains the complete problem set from which #4 is as shown above.]
I am trying to find a series representation for the following expression
$$\int_{i=0}^\infty {x^{\frac{2n-1}{2}}(b+x)^{-n}}e^{\left(-{\frac{x^2}{2m}}+\frac{x}{p}\right)} dx$$ ; b,m,n,p are constant.
Is there a name for this function?
I found a series representation for $$\int_{i=0}^\infty...
Homework Statement
i) What is the Taylor Series for f(x) = (1+x)^m about x=0 where m is a real number?
ii) Why does this binomial series terminate when m is a non-negative integer? A
iii) Can the result to (i) be used to find the first four non-zero terms of the series for (1+x)^(-1/2)...
Homework Statement
Find ∫qk(x) dx where the upper bound is 1 and the lower bound is 0. g is some function and we are finding for k = 2,6,10 and 14, hence the first four non-zero terms of a series that can be used to calculate approximations to I = ∫sin(x^2) dx were the upper bound is 1 and the...
https://drive.google.com/file/d/0B0NXDy0RMDe7MXhMcjZBdkhoSDg/view?usp=sharing
PIC: [https://drive.google.com/file/d/0B0NXDy0RMDe7MXhMcjZBdkhoSDg/view?usp=sharing]
1.
A rotating disk is connected with two arms AD and DB which are rotating with the rate of 0.2 rad/s^2 and -0.3 rad/s^2...
Hello Everyone !
I am interesting to find descriptions of the series of experiments that Newton made for determining the laws of motion. In English of course.
Homework Statement
Find the Taylor series for:
ln[(x - h2) / (x + h2)]
Homework Equations
f(x+h) =∑nk=0 f(k)(x) * hk / k! + En + 1
where En + 1 = f(n + 1)(ξ) * hn + 1 / (n + 1)!
The Attempt at a Solution
ln[(x - h2) / (x + h2)] = ln(x-h2) - ln(x + h2)
This is as far as I have been able to...
I'm trying to make an approximation to a series I'm generating; the series is constructed as follows:
Term 1:
\left[\frac{cos(x/2)}{cos(y/2)}\right]
Term 2:
\left[\frac{cos(x/2)}{cos(y/2)}-\frac{sin(x/2)}{sin(y/2)}\right]
I'm not sure yet if the series repeats itself or forms a pattern...
Homework Statement
One type of tuning circuit used in radio receivers is a series LCR circuit. You like listening to a station1 that transmits 99.3 MHz in . The government wants to make 100.1 MHz available to station2. Assume that the transmitters of the two stations are equally powerful and...
Hi,
First of all, I want to say that I know how can define and calculate Fourier coefficients but I have some question about the final presentation of Fourier and half-period or unknown period functions.
1)In this function how can we define T?
2)for above diagram, in a book, they define f(t)...
Homework Statement
f(x)=x on [0,2)
Homework Equations
Fourier Series is given as:
f(x)=a0/2 + n=1∞∑(an*cos(nπx/L) + bn*sin(nπx/L)
a0=1/L*-LL∫f(x)dxThe Attempt at a Solution
Basically what I am being taught is that we take the Period, T, to be equal to 2L so, T=2L
In this case T=2 and L=1. My...
Hello I'm working through a book (with answers) but am struggling with voltage, current, resistance and circuits. Please check my understanding below and let me know if I've finally understood. Thank you.
In particular I'm confused in Q2
Q1. A student connects light bulbs, A and B, and...
Homework Statement
Consider a circuit that consists on a resistor of an intrinsic semiconductor R and a capacitor C in series. The voltage between the terminals of the circuit is U, which is an alternated sinusoidal voltage.
U1, which is the voltage in the capacitor as a phase difference of 30...
Is it ok to assume that the entropy ##S## of an arbritary system can be written as a power series as a function of the system's internal energy ##U##? Like
$$S(U) = \sum_{i=1}^{\infty}a_iU^i = a_1 U + a_2 U^2 + \ ...$$ with ##a_i \in \mathbb{R}##.
What results could be obtained from such...
Homework Statement
Given ##b_n = 1 / n## if ##n## odd and ##b_n = 1 / n^2## if ##n## even, show that the series $$\sum_{n=1}^{\infty} (-1)^n b_n$$ diverges.
Homework Equations
Did'nt find any for this problem
The Attempt at a Solution
I assumed that ##\sum_{n=1}^{\infty} (-1)^n b_n =...
Homework Statement
(see my attached photo to better understand where I am coming from!)
So after some research, I've discovered that the current at different points in a simple series circuit is supposed to be the same value, and that the voltage is supposed to be different values.
I...
Hi guys!
(see my attached photo to better understand where I am coming from!)
So after some research, I've discovered that the current at different points in a simple series circuit is supposed to be the same value, and that the voltage is supposed to be different values.
I performed a lab on...
Homework Statement
Assume that ##a_k > 0## and ##\sum_{k=0}^\infty a_n## converges. Then for every ##\epsilon > 0##, there exists a ##n \in Bbb{N}## such that ##\sum_{k=n+1}^\infty a_k < \epsilon##.
Homework EquationsThe Attempt at a Solution
Since the series converges, the sequence of...
Suppose we have monthly totals of observed data for last 35 years. That data is of inflow of a river in a reservoir and monthly demands from the reservoir. We are interested to check the effect of construction of a dam in the upstream. The effect is, whether the downstream reservoir will have...
Homework Statement
In Complex Fourier series, how to determine the function is odd or even or neither, as in the given equation
$$ I(t)= \pi + \sum_{n=-\infty}^\infty \frac j n e^{jnt} $$Homework Equations
##Co=\pi##
##\frac {ao} 2 = \pi##
##Cn=\frac j n##
##C_{-n}= \frac {-j} n ##
##an=0##...
Homework Statement
Homework Equations
The Attempt at a Solution
a0=4
an=8/Pi*n
Heres a written solution
https://gyazo.com/57e11d1e7a360914db8aec167beb6b39.png
I watched Hellraiser: Bloodlines last night about the box that can open the gateway to hell. This is very interesting series and lines of movies. Anything similar to it you recalled? Scientists explore concepts of other multiverses, other brane dimensions now and it would be thrilling if...
It has been defined that for an alternating series, the difference between the total sum of the series and the partial sum of the series through nth term is always less than or equal to the (n+1)th term. Can anyone explain the intuitive reason behind this?
I am on book IV Wizard and Glass. Anyone else read these? I am interested in seeing The Dark Tower movie. It's one of the few science fiction series that has got my interest.
Homework Statement
Q:/ Find the complex form of Fourier series for the following periodic function whose definition in one period is given below then convert to real trigonometry also find f(0).
f(t)=cos(t/2), notes: (T=2*pi) (L=pi)
Homework Equations
1) f(t)=sum from -inf to +inf (Cn...
Hi guys, how do I figure out whether components are in series/parallel with each other just by looking at them? for example, in the circuit on the picture I posted, how do I know which of the resistors are in series/ parallel with each other? I know the definition involves nodes, but in clear...
Homework Statement
Show that the series ##\displaystyle \sum_0^{\infty} (-1)^nn## diverges
Homework EquationsThe Attempt at a Solution
It seems obvious that it diverges, for although the terms oscillate, they get bigger and bigger and never really cancel each other out. However, I am not sure...
I have a hard time comprehending excess charge in capacitor plates.
Let's say we have two identical capacitor and we charged them to identical amount of charge.
Next we connect them in series by opposite sign electrodes and we get double the voltage on the open ends.
Now a single capacitor has...
Hello and thank you for trying to help.
In spite of the fact that this seems a very simple problem, I do not find myself able to get a solution. Here it goes:
Let $$f(x)=\displaystyle \sum_{k=3}^\infty a_k \frac{x^k}{k(k-1)(k-2)}$$ and $$g(x)=\displaystyle \sum_{k=0}^\infty a_k x^k$$. Express...
i really need help about shunt and series resistances
i just want to know if they are constant or variable ?
if they are variable what make them increase or decrease?
Hiya everyone,
Alright ?
I have a simple theoretical question. In a decreasing geometric series, is it true to say that the ratio q has to be 0<q<1, assuming that all members of the series are positive ? What if they weren't all positive ?
Thank you in advance !
According to this page: https://en.wikipedia.org/wiki/Cantor's_theorem
It says: "Cantor's theorem is a fundamental result that states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself."
Furthermore, it says: "Cantor's...
Homework Statement
Write the Taylor's series expansion of the function f(x,y) = e-x2 sin(y) about the point (1,3). (The question doesn't specify how accurate it wants the answer to be, but based on the answer I have, it seems to me that the Taylor's polynomial should be of degree 3.)...
Homework Statement
Trying to find the sum of (-1)3n+1/(2n-1)3. by using term-by-term integration on the cosine Fourier series x= L/2-4L/π2∑cos(((2n-1)πx)/L)/(2n-1)2.
Homework Equations
Shown below
The Attempt at a Solution
When integrating and substituting Lx/2 for x's sine Fourier series I...
Hi,
I have a couple of questions regarding re-wiring a three phase transformer.
Say you have a three phase TX with a ratio of 415:22 and you want to use it as a single phase step-up TX (excite it with a variac or something), this is a thought-experiment.
If you excited only the middle leg of...