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
Find the sum of ##\sum\limits_{n=2}^{\infty}\ln\left(1-\dfrac{1}{n^2}\right) ##
Homework Equations
No one.
The Attempt at a Solution
At first I though it as a telescopic serie:
##\sum\limits_{n=2}^{\infty}\ln\left(1-\dfrac{1}{n^2}\right) =\ln\left(\dfrac{3}{4}\right) +...
Homework Statement
By finding a closed formula for the nth partial sum ##s_n##,
show that the series ## s=\sum\limits_{n=1}^{\infty}nx^n## converges to ##\dfrac{x}{(1-x)^2}## when ##|x|<1## and diverges otherwise.
Homework Equations
Maybe ##s=\sum\limits_{n=0}^{\infty}x^n=\dfrac{1}{1-x}## when...
Hi All
Been investigating lately ways to sum ordinarily divergent series. Looked into Cesaro and Abel summation, but since if a series is Abel Mable it is also Cesaro sumable, but no, conversely,haven't worried about Cesaro Summation. Noticed Abel summation is really a regularization...
I will state the problem below. I don't quite understand what I am needing to show. Could someone point me in the right direction? I would greatly appreciate it.
Problem:
Let p be a natural number greater than 1, and x a real number, 0<x<1. Show that there is a sequence $(a_n)$ of integers...
I've read that, in general, the energy of a wave, as opposed to what's commonly taught, isn't strictly related to the square of the amplitude. It can be seen to be related to a Taylor series, where E = ao + a1 A + a2A2 ... Also, that the energy doesn't depend on phase, so only even terms will...
Homework Statement
Determine whether the series ## \frac {(n^3+3n)^{1/2}} {5n^3+3n^2+2 sin (n)}## converges or notHomework EquationsThe Attempt at a Solution
looking at ## 1/sin (n) ## by comparison,
##1/n^2=1+1/4+1/9+1/16+...## converges for ##n≥1##
for ##n≥1 ##
implying that ##{sin (n)}≤n ##...
While I was was numerically integrating the magnetic field caused by an infinite array of magnetic moments, I observed the interesting limit ( limit (1) in the image). It may seem difficult to prove it mathematically but from the physics point of view, I think it can be proved relatively...
I am reading Paul E. Bland's book, "Rings and Their Modules".
I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.14 ... ...
Proposition 4.2.14 reads as follows...
Homework Statement
One 18 watt lamp and two 60-watt light bulb are plugged into a 120V circuit. For either DC or AC, the two bulbs are connected each other in parallel and in series with the lamp in the same circuit. Calculate;
i. the current flow through each light
ii. the total...
Hello,
I've been googling about this topic and have read from a number of different books but I still haven't found an exact answer to my question.
It is known that the current is equal when the resistors are in a series. But the resistors per definition reduce current flow. If there are 2...
I am trying to solve an integral that has ##\frac{1}{1+x^2}## as a factor in the integrand. In my book it is claimed that if we use ##\displaystyle \frac{1}{1+x^2} = \sum_{n=0}^{\infty} (-1)^n \frac{1}{x^{2n+2}}## the problem can be solved immediately. But, I am confused as to where this series...
Homework Statement
##\displaystyle \int_0^1 \frac{\arctan x}{x}dx##
Homework EquationsThe Attempt at a Solution
I converted the integral to the following; ##\displaystyle \int_0^1 \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{2n+1}dx##. In this case am I allowed to swap the summation and integral signs?
Homework Statement
Hello,
I suspect this is an easy answer but I am not seeing it. I am reviewing (more so for fun / hobby) some differential equations – I’m not in school.
I’m needing help with an example problem in Differential Equations With Boundary-Value Problems Zill 2nd edition. In...
I have the following series that I came up with in doing a problem: ##\displaystyle \sum_{n=0}^{\infty} \frac{1}{2^{n+1}(n+1)}##. I looked at WolframAlpha and it says that this series converges to ##\log (2)##. Is it possible to figure this out analytically?
Thank you for your time and effort. It is much appreciated.
1. Homework Statement
I have attached the problem with the solution to this thread. Basically, the problem asks to construct the circuit model for a generic device by using the data of terminal current and voltage measurements. From...
Homework Statement
The problem is attached as pic
Homework Equations
∑(ƒ^(n)(a)(x-a)^n)n! (This is the taylor series formula about point x = 3)The Attempt at a Solution
So I realized that we should be looking at either the 30th,31st term of the series to determine the coefficient. After we...
Homework Statement
I am looking at the wikipedia proof of uniqueness of laurent series:
https://en.wikipedia.org/wiki/Laurent_seriesHomework Equations
look above or belowThe Attempt at a Solution
I just don't know what the indentity used before the bottom line is, I've never seen it before...
Hi PF!
One way to solve a simple eigenvalue problem like
$$y''(x)+\lambda y(x) = 0,\\
y(0)=y(1)=0$$
(I realize the solution's amplitude can be however large, but my point here is not to focus on that) is to solve the inverse problem. If we say ##A[u(x)] \equiv d^2_x u(x)## and ##B[u(x)] \equiv...
Homework Statement
I'm asked to find a combination of resistors (parallel and/or series) that uses resistors of 25 Ω, 100 Ω, 50 Ω, and 50 Ω. They should add up to give a total resistance of 62.5 Ω.
Homework Equations
Req for parallel = 1/R1 + 1/R2 + ...
Req for series = R1 + R2 + ...
The...
Homework Statement
in title
Homework EquationsThe Attempt at a Solution
so i know that i have to use the ratio test but i just got completely stuck
((2x)n+1/(n+1)) / ((2x)n) / n )
((2x)n+1 * n) / ((2x)n) * ( n+1) )
((2x)n*(n)) / ((2x)1) * (n+1) )
now i take the limit at inf? i am stuck here i...
Homework Statement
Homework EquationsThe Attempt at a Solution
I don't understand why for the first part where the series goes up until arn-1, it cannot just go up until arn.. why will that first series always go up until arn-1 until it is multiplied by r?
Homework Statement
What is the power series for the function ln (x+1)? How do you find the sum of an infinite power series?
Homework Equations
sigma from n=1 to infinity (-1)^n+1 (1/n2^n)
That is the power series, how is that equivalent to ln (x+1)?
How do you find the sum, or what does it...
Hi,
I am a beginner and I don't speak very well... So I'm really sorry for my poor scientific language...
I work on 1-Dimension time series of a same system measured at different periods. In these periods, time series have different chaotic characteristics as their lyapunov exponent are...
Homework Statement
Three identical batteries are first connected in parallel to a resistor. The power dissipated by the resistor is measured to be P. After that, the batteries are connected to the same resistor in series and the dissipated power is measured to be 4P (four times larger than for...
Homework Statement
S = 1+ x/1! +x2/2! +x3/3! +...+xn/n!
To find S in simple terms.
Homework Equations
None
The Attempt at a Solution
I tried with Taylor's expansion, coshx and sinhx expansions. But cannot see consequence.
Homework Statement
Homework EquationsThe Attempt at a Solution
I am not following what is going on here, how are they getting that part that is circled. i am just completely lost here
I'm looking at the proof of the alternating series test, and the basic idea is that the odd and even partial sums converge to the same number, and that this implies that the series converges as a whole. What I don't understand is why the even and odd partial sums converging to the same limit...
Homework Statement
Homework EquationsThe Attempt at a Solution
So the book is saying that this series diverges, i have learned my lesson and have stopped doubting the authors of this book but i don't understand how this series diverges. ok i can use the comparison test using 1/3n and 1/3n...
Homework Statement
Homework EquationsThe Attempt at a Solution
So my understanding of this so far is that the whole infinite series from 1 to infinities summation minus the first six terms summation is equal to 0.0002..? This is so confusing. So how does that mean that the sum will lie...
I don't understand something, the sum n=1 until infinity of (1/n) is a divergent harmonic series meaning that its sum is infinite right?
After reading that i started thinking about the finite volume of the function (1/x) being revolved around the x-axis referred to as "Gabriels horn". They say...
Homework Statement
##f(x)=\sum_{n=0}^\infty x^n##
##g(x)=\sum_{n=253}^\infty x^n##
The radius of convergence of both is 1.
## \lim_{N \rightarrow +\infty} \sum_{n=0}^N x^n - \sum_{n=253}^N x^n##
2. The attempt at a solution
I got:
## \frac {x^{253}} {x-1}+\frac 1 {1-x}## for ##|x| \lt 1##...
Homework Statement
[/B]
Differentiate the power series for ##\frac 1 {1-x}## to find the power series for ##\frac 1 {(1-x)^2}##
(Note the summation index starts at n = 1)
2. The attempt at a solution
##\sum_{n=1}^\infty n*x^{n-1}##
Homework Statement
Homework Equations
-
The Attempt at a Solution
Here's my work :
However , the correct answer is :
Can anyone tell me where's my mistake ?
i have used series solutions to differential equations many times but i never really stopped to think why it works i understand that the series solution approximates the solution at a local provided there is no singularity in which frobenius is used but i am not understanding how exactly it...
Homework Statement
Find the Taylor expansion up to four order of x^x around x=1.
Homework EquationsThe Attempt at a Solution
I first tried doing this by brute force (evaluating f(1), f'(1), f''(1), etc.), but this become too cumbersome after the first derivative. I then tried writing: $$x^x =...
Homework Statement
I have series
\sum_{n=1}^\infty (1/n)(2^n)(-1/2)^n
Homework EquationsThe Attempt at a Solution
So trying to do the solution
(1/n)(2^n)(-\frac {1^n}{2^n})
since 1^n is going to be one for all values of n, can I say,
(1/n)(2^n)(-\frac {1}{2^n})
then...
Homework Statement
Use the power series method to solve the initial value problem:
##(x^2 +1)y'' - 6xy' + 12y = 0, y(0) = 1, y'(0) = 1##
Homework EquationsThe Attempt at a Solution
The trouble here is that after the process above I end up with ##c_{k+2} = -...
When using the method of differences on a given series, when do you stop listing the terms?
Example question:
f(r)= ; r∈N
State f(r)-f(r+1) in terms of r and hence determine
So skipping until the worked answer gives
Great so here I included the n+1th term because I'm guessing since the...
Homework Statement
Homework Equations
The Attempt at a Solution
I am stuck trying to figure out why there are three different alphas and why in the equation we are supposed to use has a theta and what that means. If I can set up the Fourier series I can properley I know how to solve it for...
Given that the sum of the first n terms of series, s, is 9-32-n
show that the s is a geometric progression.
Do I use the formula an = ar n-1? And if so, how do I apply it?
Hey! :o
I want to show that series $$f(x)=\sum_{k=1}^{\infty}2^k\sin (3^{-k}x)$$ is continuously differentiable. We have that $|2^k\sin (3^{-k}x)|\leq 2^k\cdot 3^{-k}=\left (\frac{2}{3}\right )^k$, or not?
The sum $\sum_{k=1}^{\infty}\left (\frac{2}{3}\right )^k$ converges as a geometric...
Homework Statement
Show that $$\frac{(-1)^nn!}{z^n}$$ is divergent.
Homework Equations
We can use the ratio test, which states that if, $$\lim_{n\to\infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|>1$$ a series is divergent.
The Attempt at a Solution
Applying the ratio test, we find that...
Homework Statement
Homework Equations
All I know is the a's have something to do with the integrals.
The Attempt at a Solution
I used FFT analysis in Matlab but I do not know what I am looking for. How do the a0s relate to the f(t) in the question and how would I even do run that equation in...
Homework Statement
http://sites.math.rutgers.edu/~ds965/temp.pdf (NUMBER 2)[/B]Homework Equations
I do not understand the alternating part for the second problem and the recursive part for the first problem.The Attempt at a Solution
The first answer I got was first by writing out the...