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).
Summary: Series RLC and Parallel RLC circuits
How can the voltage across a capacitor or inductor in a series RLC circuit be greater than the applied AC source voltage? The formula suggest that either can be larger than the source voltage but I still find it counter intuitive.
Also for...
Hello, I'm trying to solve this, any idea please?
Basically: Demonstrate for the next three processes if the Time Series would be stationary, if not, it should establish the conditions for it to be stationary.
Thanks
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with the proof of Theorem 2.3.9 (a) ...
Theorem 2.3.9 reads as follows:
Now, we can prove Theorem 2.3.9 (a) using the Cauchy...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help In order to formulate a rigorous proof to the proposition stated in Exercise 2.3.10 (1) ... ...
Exercise 2.3.10 (1) reads as...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with an aspect of the proof of Proposition 2.3.8 ...
Proposition 2.3.8 and its proof read as follows:
In the above proof by...
I think ##\lim_{n\rightarrow \infty} a_n = 0## since by direct substitution the value of limit won't be equal to 2 so by direct substitution we must get indeterminate form.
Then how to check for ##\sum_{n=1}^\infty a_n##? I don't think divergence test, integral test, comparison test, limit...
Because the Taylor series centered at 0, it is same as Maclaurin series. My attempts:
1st attempt
\begin{align}
\frac{1}{1-x} = \sum_{n=0}^\infty x^n\\
\\
\frac{1}{x} = \frac{1}{1-(1-x)} = \sum_{n=0}^\infty (1-x)^n\\
\\
\frac{1}{x^2} = \sum_{n=0}^\infty (1-x^2)^n\\
\\
\frac{1}{(2-x)^2} =...
Hi,
A function which could be represented using Fourier series should be periodic and bounded. I'd say that the function should also integrate to zero over its period ignoring the DC component.
For many functions area from -π to 0 cancels out the area from 0 to π. For example, Fourier series...
I enjoyed a lot the three first volumes of Zeidler's planned series of 6 books on QFT. Unfortunately, he passed away too soon.
However, it is clear from reading the first three books, that an outline of the next books in the series was already planned.
Is there a draft, containing the basis of...
In the 1960s there was a series of science-themed novels for middle school children. Each book began with a boy accidentally meeting a scientist or science-club member and being introduced to the subject. I remember one about rockets, one about geology (and one about archeology?). I don't...
Does anyone watch the Chernobyl tv series? https://www.imdb.com/title/tt7366338/
It has such realism you would think it's right in the scene.
Imdb rated it 9.6 out 10.
Did all those details in the series actualy happened? Which part is dramatization?
It made me think deep into the night...
The error ##e_{n}(y)## for ##\frac{1}{1-y}## is given by ##\frac{1}{(1-c)^{n+2}}y^{n+1}##. It follows that
##\frac{1}{1+y^2}=t_n(-y^2)+e_n(-y^2)##
where ##t_n(y)## is the Taylor polynomial of ##\frac{1}{1-y}##. Taking the definite integral from 0 to ##x## on both sides yields that...
Hi.
The #1 complaint about the Greiner series are the typos. It's a shame how it undermines the potential of this series which is, otherwise, highly acclaimed. Do you know where errata for any of these volumes can be found? Or perhaps those who have read some of the books could share their own...
I'm not too sure how to use the hint here. What I had so far was this: an odd extension of ##f## implies ##f = \sum_{k=1}^\infty b_k \sin(k x)##. Notice for ##m>n## $$ \left|\sum_{k=1}^m b_k\sin(k x) - \sum_{k=1}^n b_k\sin(k x)\right| = \left| \sum_{k=n+1}^m b_k\sin(k x)\right| \leq...
The problem of the interaction of a point charge with a dielectric plate of finite thickness implies the existence of an infinite series of image charges (see http://www.lorentzcenter.nl/lc/web/2011/466/problems/2/Sometani00.pdf). I introduce notations identical to those used in this work. The...
Hi, I have some crucial questions belong to statistics:
First, How can we derive the variance function with respect to mean for a given data?
Secondly, I would like to ask: what method should we employ if the variance in time series behaves like a high order (such as ##𝑎𝑢_𝑡^5+𝑏𝑢_𝑡^4+𝑐𝑢_𝑡^3##...
For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.
Hi, as you know infinite sum of taylor series may not converge to its original function which means when we increase the degree of series then we may diverge more. Also you know taylor series is widely used for an approximation to vicinity of relevant point for any function. Let's think about a...
I've lately been interested in series and how they converge to interesting values. It's always interesting to see how they end up adding up to something involving pi or e or some other unexpected solution.
I learned about the Leibniz formula back in college : pi/4 = 1/1-1/3+1/5-1/7+1/9-...
and...
Hi, I've been reading the passage attached below and from what I understand we are looking at a 1D chain of atoms and if anyone atom moves it changes the potential for surrounding atoms and cause a change in energy in the system so the total energy is dependent on all the positions of the atoms...
I have already solved up to after the switches are flipped, and all the charge is on C1. See the second attached image for a detailed diagram of the situation after the switches are flipped. However, the notes then say that all the charge is trapped between C1 and C2, which I don't understand...
I am attempting to find the sine representation of cos 2x by letting
$$f(x) = \cos2x, x>0$$ and $$-\cos2x, x<0$$
Which is an odd function. Hence using $$b_n = \dfrac{2}{l} \int^\pi _0 f(x) \sin(\dfrac{n\pi x}{l})dx$$ I obtain $$b_n = \dfrac{2n}{\pi} \left( \dfrac{(-1)^n - 1}{4-n^2} \right)$$...
Given that they're all on the same branch, I had assumed that they were in series with one another. But with the middle resistor having being on the middle of three branches, it looks parallel.
Like I said, I have a feeling it's in series (making the answer 3R).
This question is from a past...
Hey all, it's been awhile since done any calculus or DE's but was trying out some modelling (best price price per item for bulk value deals as a function of time and amount), in the last line i have f(n,t) implicitly.
Any pointers or techniques for solving such things?
How do you increase torque in gerotor design other than increasing flow. Will lengthening it increase torque? Will increasing diameter increase torque? What about running 2 or 3 in series?
Curious about proving that ##\sum_{m=2}^\infty \sum_{n=2}^\infty 1/n^m ## = 1
ran this in Matlab and n,m to 2:1000 =0.9990, and n,m 2:10000 =0.9999, so it does appear to converge to 1
Homework Statement
Hello,
i am trying to do find the Fourier series of abs(sin(x)), but have some problems. As the function is even, bn = 0. I have calculated a0, and I am now working on calculating an. However, when looking at the solution manual, they have set up one calculation for n > 1...
Homework Statement
I have encountered this problem from the book The Physics of Waves and in the end of chapter six, it asks me to prove the following identity as part of the operation to prove that as the limit of ##W## tends to infinity, the series becomes an integral. The series involved is...
Homework Statement
Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n##
Evaluate: ##f^{(8)}(4)##
Homework Equations
The Taylor Series Equation
The Attempt at a Solution
Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except for the...
Hi all. Could someone work out for me how equation 21 in attachment left side becomes right side. Please show in detail if you could.
It's for exponential Fourier series.
Drforbin
thank you
Homework Statement
Test the following series for convergence or divergence.
##\sum_{n = 1}^{\infty} \frac {\sqrt n} {e^\sqrt n}##
Homework Equations
None that I'm aware of.
The Attempt at a Solution
I know I can use the Integral Test for this, but I was hoping for a simpler way.
Homework Statement
Determine the series that is equal to the integral ##\int_0^1 x^2\cos(x^3)dx##
Homework EquationsThe Attempt at a Solution
So I didn't really know what I was doing but I did end up with the correct solution.
What I did was to find a Taylor Series for the integrand, this...
##\sum_{n=1}^\infty 1/n^2 ## converges to ##π^2/6##
and every other series with n to a power greater than 1 for n∈ℕ convergesis it known if the sum of all these series - ##\sum_{m=2}^\infty \sum_{n=1}^\infty 1/n^m ## for n∈ℕ converges?
apologies for any notational flaws
Homework Statement
Hello. I'm not entirely sure what this question is asking me, so I'll post it and let you know my thoughts, and any input is greatly appreciated.
If the series ##\sum_{n=0}^\infty a_n(x-4)^n## converges at x=6, determine if each of the intervals shown below is a possible...
Reading this piece with a number of proofs of the divergence of the harmonic series
http://scipp.ucsc.edu/~haber/archives/physics116A10/harmapa.pdf
and this example states: 'While not completely rigorous, this proof is thought-provoking nonetheless. It may provide a good exercise for students...
One can have a progression and it is called a Sequence.
One can sum the terms in a sequence or progression, and this is called a Series.
Why those terms like that; or why those two different terminologies? Was it decided just to pick a word Series so as to avoid the need to use Sum Of the...
I've seen the proof that the sum of 1/n for = 1 to infinity is infinity (which still blows my mind a little).
Is the sum of 1/nn for n = 1 to infinity also infinity?
i.e, 1 + 2/4 + 3/27 + 4/256+...
Homework Statement
Use the integral test to compare the series to an appropriate improper integral, then use a comparison test to show the integral converges or diverges and conclude whether the initial series converges or diverges.
##\sum_{n=3}^\infty \frac{n^2+3}{n^{5/2}+n^2+n+1}##
Homework...
Homework Statement
Obtain Maclaurin Series for:
f(x) = sin(x2)/x
Homework Equations
f(x) = ∑f(n)(c) (x-c)n / n! (for Maclaurin c = 0)
The Attempt at a Solution
I know that sin(x2) = x2 - (x2*3/3! +...
from the final answer I see, that this is just multiplied to 1/x.
This bothers me...
Homework Statement
- Given a bounded sequence ##(y_n)_n## in ##\mathbb{C}##. Show that for every sequence ##(x_n)_n## in ##\mathbb{C}## for which the series ##\sum_n x_n## converges absolutely, that also the series ##\sum_n \left(x_ny_n\right)## converges absolutely.
- Suppose ##(y_n)_n## is...
I know this may sound as a stupid question but I would like to clarify this.
An arbitrary function f can be expressed in the Fourier base of sines and cosines. My question is, Can this method be used to solve any differential equation?
You plug into the unkown function the infinite series and...
Hi,
I've computed 512 terms of a power series numerically. Below are the first 20 terms.
$$
\begin{align*}
w(z)&=0.182456 -0.00505418 z+0.323581 z^2-0.708205 z^3-0.861668 z^4+0.83326 z^5+0.994182 z^6 \\ &-1.18398 z^7-0.849919 z^8+2.58123 z^9-0.487307 z^{10}-7.57713 z^{11}+3.91376 z^{12}\\...
Hi Everyone,
I am currently working on a project where I am creating English subtitles for an Italian TV Series, for deaf and hard of hearing. There is one particular line I am struggling to hear and it is where one of the characters is talking about a type of physics. I wondered if anyone...
Homework Statement
The battery is disconnected from a series RC circuit after the capacitor is fully charged and is replaced by an open switch. When the switch is closed,
a. the capacitor does not allow current to pass
b. the current stops in the resistor
c. the potential difference across the...