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).
a) I think I got this one (I have to thank samalkhaiat and PeroK for helping me with the training in LTs :) )
$$\eta_{\mu\nu}\Big(\delta^{\mu}_{\rho} + \epsilon^{\mu}_{ \ \ \rho} +\frac{1}{2!} \epsilon^{\mu}_{ \ \ \lambda}\epsilon^{\lambda}_{ \ \ \rho}+ \ ...\Big)\Big(\delta^{\nu}_{\sigma} +...
Here is a circuit diagram:
.
We have three capacitors, with capacitances ##C_1##, ##C_2## and ##C_3##. Plates are labelled as ##A_1, A_2, A_3 ... A_6##. Point P is connected to the positive terminal of the battery and point N is connected to the negative terminal of the...
Known: V source = 30.0 V
, R1 = 15.0 W, R2 = 15.0 W, R3 = 15.0 W
To determine the current, first find the equivalent resistance.
I = Vsource/R and R = RA + RB
= Vsource/RA + RB
30.0 V/15.0 W + 15.0 W + 15.0 W
= 1.5 A
This is as far as I could do the work for this question. I’m having trouble..
Hi All,
I've been going through Shankar's 'Principles of Quantum Mechanics' and I don't quite understand the point the author is trying to make in this exercise. I get that this wavefunction is not a solution to the Schrodinger equation as it is not continuous at the boundaries and neither is...
I recently stumbled upon Gradshteyn, Ryzhik: Table of Integrals, Series, and Products
and it is worth recommending for all who have to deal with actual solutions, i.e. especially engineers, physicists and all who are confronted with calculating integrals, series and products.
Summary:: If ##f(x)=-f(x+L/2)##, where L is the period of the periodic function ##f(x)##, then the coefficient of the even term of its Fourier series is zero. Hint: we can use the shifting property of the Fourier transform.
So here's my attempt to this problem so far...
Mentor note: Moved from technical section, so is missing the homework template.
Im doing some older exams that my professor has provided, but I haven't got the solutions for these. Can someone help confirm that the solutions I've arrived at are correct?
Modern batteries use double-sided anode and cathodes for greater energy density. Series wiring of batteries is typically accomplished by connecting the anode of one cell to the cathode of another. However, can series be accomplished by stacking double-sided anode and cathode alternatingly with...
since the first term is ##g(0)= \frac {1}{3}##
& last term is ##g(1)=\frac {4}{6}##
it follows that the ##\sum_{0}^1 g(x)##= ##\frac {1}{3}##+##\frac {4}{6}=1## is this correct?
Hello.
I have completed the following question.
My answer:
i)
Circuit Impedance
Reactance = XL = 2 x pi x F x L
= 2 x pi x 50 x 0.15
= 47.12 Ohms
Reactance of Capacitor = XC = 1/2 x pi x F x C...
Dear Everyone,
I am having trouble with finding a formula of the multiplication 3 formula power series.
\[ \left(\sum_{n=0}^{\infty} a_nx^n \right)\left(\sum_{k=0}^{\infty} b_kx^k \right)\left(\sum_{m=0}^{\infty} c_mx^m \right) \]
Work:
For the constant term:
$a_0b_0c_0$
For The linear...
Set ##\epsilon=\frac{1}{2}##. Let ##N\in \mathbb{N}## and choose ##n=N,m=2N##. Then:
##\begin{align*}
\left|s_N-s_{2N}\right|&=&\left|\sum_{l=1}^N \frac{1}{l} - \sum_{l=1}^{2N} \frac{1}{l}\right|\\...
Hi all,
In an LRC AC series circuit, at which frequencies are where you are mainly dumping your generator/current energy into capacitor to create electric fields or into the inductor to create magnetic fields? So, for example, at low frequencies, f --> 0, the impedance of the inductor goes to 0...
Not really a homework problem, just an equation from my textbook that I do not understand. I can't think of any way to even begin manipulating the right hand side to make it equal the left hand side.
Just to confirm equality (thanks to another user for suggestion), I multiplied both sides by of...
LIGO India EPO (Education and Public Outreach) team is hosting a series of talks on Youtube. No registration or any formalities; just tune into the LIGO India EPO Youtube channel and you can attend the lectures.
Following is the list of upcoming talks:
20th April: Speaker: Prof. Ajith...
Interestingly, If I neglect the ##(-1)^n## or ##(-1)^{n+1}## then apply preliminary test, I could find the limit. Whether the limit is not equal to zero, as in series number 1 and 2, then I can conclude the series is divergent. But, if the limit is equal to zero, as in series number 3, then I...
1. Is it because the initial formula start the series from ##n = 2##?
2. If the initial formula is used, can I find ##S##, which $$S=\lim_{n\to\infty} \frac{2}{n^2-1}=\frac{2}{\infty}=0$$? Why that answer is different if the formula is changed.
I have used ##\sim## but meant ##\sum_{k=0}^\infty##
my math homework platform is telling me that this is wrong. I've tried using desmos to test it and it was a perfect match. Any ideas on where I went wrong?
This is a second grade high school problem, translated from my native language.
I don't have a problem with calculating, but with understanding the concept. There is an instruction with the assignment that says: The capacitor can be viewed as a combination of two capacitors in series with...
I am revising perturbation theory from Griffiths introduction to quantum mechanics.
Griffiths uses power series to represent the perturbation in the system due to small change in the Hamiltonian. But I see no justification for it! Other than the fact that it works.
I searched on the internet a...
https://scholar.harvard.edu/files/david-morin/files/relativity_chap_1.pdf
The questions start at page 44
Whenever I refer to y, y = gamma.
1.1
This question is primarily deriving LV/C^2?
How does 2LV / c^2-v^2 becomes
2Lv / c^2(1-v^2/c^2)1.4
On the solution page it shows fig 1.61 and fig...
I found total capacitance and inserted the total capacitance and emf of cell in equation CV =Q. However I know that there is a resistor connected so that this accounts for lost volts
I'm currently typing up some notes on topics since I have free time right now, and this question popped into my head.
Given a problem as follows:
Find the first five terms of the Taylor series about some ##x_0## and describe the largest interval containing ##x_0## in which they are analytic...
I know that the Lyman series is a series of lines in the ultra-violet. So that means a higher frequency so it will fall on the right of the diagram.
And Paschen is infrared. So a lower frequency so on the left side of the diagram?
A Harvard teams believe they have found a protein series they call "Hemolithin" in an asteroid.
Isotopes and other evidence indicates that it is not from a terrestrial source."Astrobiology Web" link
arxiv pdf link
We transform the series into a power series by a change of variable:
y = √(x2+1)
We have the following after substituting:
∑(2nyn/(3n+n3))
We use the ratio test:
ρn = |(2n+1yn+1/(3n+1+(n+1)3)/(2nyn/(3n+n3)| = |(3n+n3)2y/(3n+1+(n+1)3)|
ρ = |(3∞+∞3)2y/(3∞+1+(∞+1)3)| = |2y|
|2y| < 1
|y| = 1/2...
∑((√(x2+1))n22/(3n+n3))
We use the ratio test:
ρn = |2(3n+n3)√(x2+1)/(3n+1+(n+1)3)|
ρ = |2√(x2+1)|
ρ < 1
|2√(x2+1)| < 1
No "x" satisfies this expression, so I conclude the series doesn't converge for any "x". However the answer in the book says the series converges for |x| < √(5)/2. What am...
∑(x2n/(2nn2))
We use the ratio test:
ρn = |(x2n2/(2(n+1)2)|
ρ = |x2/2|
ρ < 1
|x2| < 2
|x| = √(2)
We investigate the endpoints:
x = 2:
∑(4n/(2nn2) = ∑(2n/n2))
We use the preliminary test:
limn→∞ 2n/n2 = ∞
Since the numerator is greater than the denominator. Therefore, x = 2 shouldn't be...
My partner asked me about questions no. 8 and 9.
Number 8 asks about what is the area of the quadrilateral.
Number 9 asks about what number is below the number 25.
Those are questions for Elementary School Math Olympiads in my country but both of us were having a hard time figuring them out...
Moved from technical forum section, so missing the homework template
Let x be a real number. Find the function whose power series is represented as follows: x^3/3! + x^9/9! + x^15/15! ...
I see that there is a connection to the power series expansion of e^x but am having difficulty finding...
Suppose I have a galvanic cell, where I've arbitrarily set the (-) anode to have a potential of zero volts and the (+) cathode to ##\epsilon## V. The electrodes are connected via the load, but also via the solutions and salt bridge in the centre. Edit: The two trailing wires are connected to a...
At the exam i had this power series
but couldn't solve it
##\sum_{k=0}^\infty (-1)^\left(k+1\right) \frac {k} {log(k+1)} (2x-1)^k##
i did apply the ratio test (lets put aside for the moment (2x-1)^k ) to the series ##\sum_{k=0}^\infty \frac {k} {log(k+1)}## in order to see to what this...
Summary:: Trying to find Rth but I do not get the same value as the one from the solution.
[moderator: moved from a technical forum. No template.]
I am trying to find Rth to solve this problem, however once I simplified it, I get a value of 700.745 Ω while in the solution, the answer is...
Sorry if the answer is obvious, but I was wondering if positive and negative electrodes (cells in series) can share the same current collector as depicted below? I want to create a 12V battery with cells inline in series without creating cells with individual current collectors. Note that the...
##\sum_{k=0}^\infty \frac {2^n+3^n}{4^n+5^n} x^n##
in order to find the radius of convergence i apply the root test, that is
##\lim_{n \rightarrow +\infty} \sqrt [n]\frac {2^n+3^n}{4^n+5^n}##
##\lim_{n \rightarrow +\infty} \left(\frac {2^n+3^n}{4^n+5^n}\right)^\left(\frac 1 n\right)=\lim_{n...
given the following
##\sum_{n=0}^\infty n^2 x^n##
in order to find the radius of convergence i do as follows
##\lim_{n \rightarrow +\infty} \left |\sqrt [n]{n^2}\right|=1##
hence the radius of convergence is R=##\frac 1 1=1##
|x|<1
Now i have to verify how the series behaves at the...
##\sum_{n=0}^\infty (-1)^n \frac {x^\left(n+1\right)}{n+1}## for x=1
##\sum_{n=0}^\infty (-1)^n \frac {1^\left(n+1\right)}{n+1}##
i've tried leibniz test but i can only find that it converges
why is this power equal to ##log(2)##?
i've also tried with ##\sum_{n=0}^\infty\log \left (1+\frac 1...
ANY AND ALL HELP IS GREATLY APPRECIATED :smile:
I have found old posts for this question however after reading through them several times I am having a hard time knowing where to start.
I am happy with the sketch that the function is correctly drawn and is neither odd nor even. It's title is...
##\sum_{n=1}^\infty \frac {(sin 𝜶)^n}{2n} ##
I apply the root test and i get
##\lim_{n \rightarrow +\infty} \frac {sin 𝜶}{2n^\frac 1 n} ##
at this point i don't know how to treat the denominator.
## \sum_{n=1}^\infty (-1)^n \frac {log(n)}{e^n}##
i take the absolute value and consider just
## \frac {log(n)}{e^n}##
i check by computing the limit if the necessary condition for convergence is satisfied
##\lim_{n \rightarrow +\infty} \frac {log(n)}{e^n} =\lim_{n \rightarrow +\infty}...
This is written on Greiner's Classical Mechanics when solving a Tautochrone problem.
Firstly,I don’t understand why we didn’t use the term ##m=0##
and Sencondly, how the integrand helps us to fulfill the Dirichlet conditions. That means,how do we know that the period is 1?Thanks
I find this interesting. A pretty detailed description, of a complex geological series of events, that can't be directly seen.
Here's my summary:
In 2018 an usual humming was picked up by seismic equipment an island off Africa, a magma pool drained, flowed up a dyke, when horizontal, and then...