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).
First i computed the rate of energy wrt time of the second plate:
$$dq_{2}/dt = A \sigma ((373)^4/2 + (273)^4/2 - T_{2}^4)$$
Equaliting it to zero we get the answer. But i am not sure why do we need to equality it to zero.
The q arrow on the figure suggest me that it is conservation of energy...
The spectrochemical series of metals, under the circumstances that same ligands are used and that it is in an octahedral coordination, is given by:
Mn2+ < Ni2+ < Co2+ < Fe2+ < V2+ < Fe3+ < V3+ < Co3+ < Mn4+ < Mo3+ < Rh3+ < Ru3+ < Pd4+ < Ir3+ < Pt4+
When I was skimming through a textbook to...
Attached is the section from the book. I am doing section 31.3
We know that an AC source gives a sinusoidal varying current, and as far as I know its always given by ##i(t) = Icos(wt)##. Its like we take the current to be the base of all other quantities, so we use it to derive everything else...
My attempt: If ##i = 1##, then ##\gamma_1 = G \rhd G' = \gamma_2##. We proceed by induction on ##i##. Consider an element ##xyx^{-1}y^{-1}## where ##x \in \gamma_i## and ##y \in G##. Since ##\gamma_i \rhd G##, we have ##yx^{-1}y^{-1} = x_0 \in \gamma_i##. So, ##xyx^{-1}y^{-1} = xx_0 \in \gamma_i...
Hello, I am currently self studying sequence and series and I got to a topic called arithmetic-geometric sequence, and after the theory It gives this exercise:
1) Find the sum:
S=1+11+111+1111+...+111...111, if the last (number) is a digit of n.
I was given a tip That says that
1 = (10 - 1)/9...
I am lookin designing balancing resistors for series capacitors and understand that I need to consider the leakage current from the capacitors. I am trying to determine factors that would case the insulation resistance to decrease over time so I can design around that.
Very simply, I can't understand why the charges of capacitors placed in series are all the same, and why even the total one(of the circuit) is equal to those.
How is it possible that the total charge is the same as the individual ones?
There must be some concept/property about capacitors which...
Some popular math videos point out that, for example, the value of -1/12 for the divergent sum 1 + 2 + 3 + 4 ... can be found by integrating n/2(n+1) from -1 to 0. We can easily verify a similar result for the sum of k^2, k^3 and so on.
Is there an elementary way to connect this with the more...
If given a set of data points for the magnitude of a cepheid variable at a certain time (JD), how can we use Fourier series to find the period of the cepheid variable?
I'm trying to do a math investigation (IB math investigation) on finding the period of the cepheid variable M31_V1 from data...
Mary Boas attempts to explain this by pointing out that the situation cannot arise because charges will have to be placed individually, and in an order, and that order would represent the order we sum in. That at any point the unplaced infinite charges would form an infinite divergent series...
Attempt: Consider an arbitrary normal series ##G = G_0 \ge G_1 \ge G_2 \ge \dots \ge G_n = 1##. We will refine this series into a composition series. We start by adding maximal normal subgroups in between ##G_0## and ##G_1##. If ##G_0/G_1## is simple, then we don't have to do anything. Choose...
Attempt so far: We're given that ##G## and ##H## have equivalent normal series
$$G = G_0 \ge G_1 \ge \dots \ge G_n = 1$$
and
$$H = H_0 \ge H_1 \ge \dots \ge H_n = 1$$
We can assume they have the same length because they are equivalent. I think from here I need to construct two composition...
So here's my homework question:
This is the reference formula along with the Rung-Kutta form with the variables mentioned in the question
Here is my attempt so far:
Problem is that i am unsure how to expand this to even get going. I tried referencing my text Math Methods by Boas which has...
Prove
$$
\zeta(2) = \sum_{n\in \mathbb{N}}\dfrac{1}{n^2} = \dfrac{\pi^2}{6}
$$
by evaluating
$$
\int_0^1\int_0^1\dfrac{1}{1-xy}\,dx\,dy
$$
twice: via the geometric series and via the substitutions ##u=\dfrac{y+x}{2}\, , \,v=\dfrac{y-x}{2}##.
I'm trying to get from the formula in the top to the formula in the bottom (See image: Series). My approach was to complexify the sine term and then use the fact that (see image: Series 1) for the infinite sum of 1/ne^-n. Then use the identity (see image: Series 2). Any other ideas?
Would somebody be kind enough to check whether I've picked the right convergence tests for each of these and reached the right answers? There are no solutions in the book.
Also, is there a method I can use to determine if I'm right - does calculating the first n terms help?
Thank you
Edit...
Prove that $\sqrt[3]{\dfrac{2}{1}}+\sqrt[3]{\dfrac{3}{2}}+\cdots+\sqrt[3]{\dfrac{996}{995}}-\dfrac{1989}{2}<\dfrac{1}{3}+\dfrac{1}{6}+\cdots+\dfrac{1}{8961}$.
hi guys
i am trying to solve the Gaussian integral using the power series , and i am suck at some point : the idea was to use the following series :
$$\lim_{x→∞}\sum_{n=0}^∞ \frac{(-1)^{n}}{2n+1}\;x^{2n+1} = \frac{\pi}{2}$$
to evaluate the Gaussian integral as its series some how slimier ...
I was trying to find this form of the Taylor series online:
$$\vec f(\vec x+\vec a) = \sum_{n=0}^{\infty}\frac{1}{n!}(\vec a \cdot \nabla)^n\vec f(\vec x)$$
But I can’t find it anywhere. Can someone confirm it’s validity and/or provide any links which mention it? It seems quite powerful to be...
I have the following function
$$f^{(0)}\left(x\right)=f\left(x\right)=e^{x}$$
And want to approximate it using Taylor at the point ##\frac{1}{\sqrt e} ##
I also want to decide (without calculator)whether the error in the approximation is smaller than ##\frac{1}{25} ##
The Taylor polynomial is...
Hi!
Some time ago I came across a series and never solved it, I tried to give a new go because I was genuinely curious how to tackle it, which I thought would work, because it looks innocent, but there is something about the beast making it hard to approach for me. So need some help! Maybe this...
Given the circuit above, I have to solve for the labelled currents, find V total and R total accordingly. 1A is flowing through the 5Ω resistor as shown. Assuming electron flow (negative terminal to positive) for circuit.
The connector in the middle was somewhat confusing. Without it, this...
Can someone help me on this question? I'm finding a very strange probability distribution.
Question: Suppose that x_1 and x_2 are independent with x_1 ~ geometric(p) and x_2 ~ geometric (1-p). That's x_1 has geometric distribution with parameter p and x_2 has geometric distribution with...
My homework is on mathematical physics and I want to know the concept behind Laurent series. I want to know clearly know the process behind attaining the series representation for the expansion in sigma notation using the formula that can be found on the attached files. There are three questions...
Question 1; Method 1
If the sum of the first four terms is 139 then S4=139
139=1/2(4)(2a+(4-1)d)
139=2(2a+3d)
139=4a+6d----- [1]
The part of this question that is confusing is the "the sum of the next four terms is 115".
Would this mean that S8=S4+115=139+115=254?
In which case...
A beam of length L with fixed ends, has a concentrated force P applied in the center exactly in L / 2.
In the differential equation:
\[ \frac{d^4y(x)}{dx^4}=\frac{1}{\text{EI}}q(x) \]
In which
\[ q(x)= P \delta(x-\frac{L}{2}) \]
P represents an infinitely concentrated charge distribution...
The American Physical Society (APS) has just launched a webinar series addressing careers for physicists in industry (broadly includes national labs). [ETA: Correction: National labs are covered under a separate series also listed on the following link.] Details of the series can be found...
Show by contradiction that
$$
\sum_{p\in \mathbb{P}}\dfrac{1}{p} =\sum_{p\;\text{prime}}\dfrac{1}{p}
$$
diverges. Which famous result is an immediate corollary?
The sum of the harmonic series(1/1+1/2+1/3...) is infinite. However, if you exclude all the terms that contain the number nine, the sum is just under 23.
From 1 to 100 19% of the terms are excluded
From 1 to 1000 27.1% of the terms are excluded
Is there a formula for a N digit number what the...
Hi,
Question: If we have an initial condition, valid for -L \leq x \leq L :
f(x) = \frac{40x}{L} how can I utilise a know Fourier series to get to the solution without doing the integration (I know the integral isn't tricky, but still this method might help out in other situations)?
We are...
Hey! 😊
I want to check the convergence for the below series.
- $\displaystyle{\sum_{n=1}^{+\infty}\frac{\left (n!\right )^2}{\left (2n+1\right )!}4^n}$
Let $\displaystyle{a_n=\frac{\left (n!\right )^2}{\left (2n+1\right )!}\cdot 4^n}$.
Then we have that \begin{align*}a_{n+1}&=\frac{\left...
While transforming the equation of the Basel problem, the following infinite series was obtained.
$$\sum_{n=1}^{\infty} \frac{n^2+3n+1}{n^4+2n^3+n^2}=2$$
However I couldn't think of a simple way to prove that.
Can anyone prove that this equation holds true?
I read Iterative methods for optimization by C. Kelley (PDF) and I'm struggling to understand proof of
Notes on notation: S is a simplex with vertices x_1 to x_{N+1} (order matters), some edges v_j = x_j - x_1 that make matrix V = (v_1, \dots, v_n) and \sigma_+(S) = \max_j \lVert...
Hi.
Any fans of DARK here?
This show was mind blowing. The feel of the show may be to heavy for some, especially in the beginning. So it may not be your style. But if you like that atmosphere, the series is truly amazing. The writing is on a whole new level. If you haven't watched it, I...
Dear Everyone,
I am wondering how to use the integral formula for a holomorphic function at all points except a point that does not exist in function's analyticity. For instance, Let $f$ be defined as $$f(z)=\frac{z}{e^z-i}$$. $f$ is holomorphic everywhere except for $z_n=i\pi/2+2ni\pi$ for...
I just watched season one of Tales from the Loop. It has the mood of Interstellar on the Earth where people live quiet lives of desperation. There’s an underground physics lab nicknamed the Loop where the impossible becomes possible. There’s the people whose lives are affected in strange ways...
First, I try to define the function in the figure above: ##V(t)=100\left[sin(120\{pi}\right]##.
Then, I use the fact that absolute value function is an even function, so only Fourier series only contain cosine terms. In other words, ##b_n = 0##
Next, I want to determine Fourier coefficient...
Hello Everyone!
I want to learn about Fourier series (not Fourier transform), that is approximating a continuous periodic function with something like this ##a_0 \sum_{n=1}^{\infty} (a_n \cos nt + b_n \sin nt)##. I tried some videos and lecture notes that I could find with a google search but...
I tried to use the ratio test, but I am stuck on finding the range of the limit.
$$\because \left|x-1\right|<1.5=Radius$$
$$\therefore -0.5<x<2.5$$
$$\lim _{n \to \infty} \left| \frac{A_{n+1}(x-1)^{n+1}}{A_n(x-1)^n} \right|$$
$$\lim_{n \to \infty} \frac{A_{n+1} \left|x-1\right|}{A_n} <1$$...
Hi,
I was watching a video on the origin of Taylor Series shown at the bottom.
Question 1:
The following screenshot was taken at 2:06.
The following is said between 01:56 - 02:05:
Halley gives these two sets of equations for finding nth roots which we can generalize coming up with one...
Hi,
A person has 40 litres of milk. As soon as he sells half a litre, he mixes the remainder with half a litre of water. How often can he repeat the process, before the amount of milk in the mixture is 50% of the whole?
Detailed explanation is appreciated.:)
Solution:
I am working on...
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...