What is Power series: Definition and 642 Discussions
In mathematics, a power series (in one variable) is an infinite series of the form
where an represents the coefficient of the nth term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function.
In many situations c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form
Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at 1⁄10. In number theory, the concept of p-adic numbers is also closely related to that of a power series.
If ##a_n\neq 0## for all ##n##, consider the limit
$$\lim\limits_{n\to\infty} \left | \frac{a_{n+1}x^{n+1}}{a_nx^n} \right | = \lim\limits_{n\to\infty}\left |\frac{a_{n+1}}{a_n}\right | |x|=L$$
The ratio test asserts that this series converges if ##L<1## and diverges if ##L>1##.
I'm a bit...
I am attempting to solve this differential equation with power series
I came with the following solution but I doubt it is correct.
Since x=1 we get:
I doubt its correctness because it looks messy. Also the convergence radian R goes to 0, giving only a solution for x=0 which is not...
The way the formula is stated in my noname lecture notes is as follows:
Then they remark that:
The last sentence puzzles me deeply, specifically the part "When ##|z_0|<R##...". Why can we expand ##\frac1{z-z_0}## when ##|z_0|<R##? This makes little sense to me. I have noticed several typos...
I take $$P(z)=\frac {1-z}{5-z} = \frac 1 5 -\frac 4 {25} z - \frac 4 {125} z^2 - \cdots$$ which has radius of convergence 5, and $$Q(z)=\frac {5-z} {1-z} = 5+4z+4z^2+\cdots$$ which has radius of convergence 1.
##P(z)Q(z)=1## converges everywhere.
Is this correct? If so, do you think it's a good...
In Ordinary Differential Equations by Adkins and Davidson, in a chapter on the Laplace transform (specifically, in a section where they discuss the linear space ##\mathcal{E}_{q(s)}## of input functions that have Laplace transforms that can be expressed as proper rational functions with a fixed...
In these lecture notes, there is the following theorem and proof:
I'm confused about "...the power series converges if ##0\leq r<1##, or ##|x-c|<R##...". In other words, why is ##|x-c|<R## equivalent to ##0\leq r<1##?
I guess the author reasons as follows. If $$R=\lim _{n\to \infty...
I am reading the following passage in these lecture notes (chapter 10, in the proof of theorem 10.3) on power series (and have seen similar statements in other texts):
I'm confused about ##|x_0|<R##.
If ##M=\sup (A)##, then for every ##M'<M##, there exists an ##x\in A## such that ##x>M'##...
Margules suggested a power series formula for expressing the activity composition variation of a binary system.
lnγ1=α1x2+(1/2)α2x2^2+(1/3)α3x2^3+...
lnγ2=β1x1+(1/2)β2x1^2+(1/3)β3x1^3+...
Applying the Gibbs-Duhem equation with ignoring coefficients αi's and βi's higher than i=3, we can obtain...
I had difficulty showing this no matter what I tried in (a) I am not getting it. Here for p(t) in K[[t]] : ## |p|=e^{-v(p)} ## where v(p) is the minimal index with a non-zero coiefficient.
I said that p_i is a cauchy sequence so, for every epsilon>0 there exists a natural N such that for all...
I believe I am doing everything right up until the point where I have to try and find a recurrence relation. I honestly have no idea what to do from there. I've listed my work in getting the powers of n and the indicies to all match. Any help appreciated.
Here is the original DE...
Just earlier today i was practicing solving some ODEs with the power series method and when i did it to the infinite square well i noticed that my final answer for ##\psi(x)## wouldn't give me the quantised energies. My solution was
$$\psi(x) = \sum^{\infty}_{n=0} k^{2n}(\cos(x) + \sin(x))$$...
I have to find 2 solutions of this Bessel's function using a power series.
##x^2 d^2y/dx^2 + x dy/dx+ (x^2 -9/4)y = 0##
I'm using Frobenius method.
What I did so far
I put the function in the standard form and we have a singularity at x=0. Then using ##y(x) = (x-x_0)^p \sum(a_n)(x-x_0)^n##...
We usually talk about ##F[[x]]##, the set of formal power series with coefficients in ##F##, as a topological ring. But we can also view it as a topological vector space over ##F## where ##F## is endowed with the discrete topology. And viewed in this way, ##\{x^n:n\in\mathbb{N}\}## is a...
hi guys
i am trying to solve the Gaussian integral using the power series , and i am suck at some point : the idea was to use the following series :
$$\lim_{x→∞}\sum_{n=0}^∞ \frac{(-1)^{n}}{2n+1}\;x^{2n+1} = \frac{\pi}{2}$$
to evaluate the Gaussian integral as its series some how slimier ...
Hi!
Some time ago I came across a series and never solved it, I tried to give a new go because I was genuinely curious how to tackle it, which I thought would work, because it looks innocent, but there is something about the beast making it hard to approach for me. So need some help! Maybe this...
I tried to use the ratio test, but I am stuck on finding the range of the limit.
$$\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 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...
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...
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?
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...
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...
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...
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...
##\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...
I have an ODE:
(x-1)y'' + (3x-1)y' + y = 0
I need to find the solution about x=0. Since this is an ordinary point, I can use the regular power series solution.
Let y = ## \sum_{r=0}^\infty a_r x^r ##
after finding the derivatives and putting in the ODE, I have:
## \sum_{r=0}^\infty a_r...
That's my attempting: first I've wrote ##e## in terms of the power series, but then I don't how to get further than this $$ \sum_{n=0}^\infty (-1)^n \frac {Â^n} {n!} \hat B \sum_{n=0}^\infty \frac {Â^n} {n!} = \sum_{n=0}^\infty (-1)^n \frac {Â^2n} {\left( n! \right) ^2} $$. I've alread tried to...
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...
Homework Statement
This is from a complex analysis course:
Find radius of convergence of
$$\sum_{}^{} (log(n+1) - log (n)) z^n$$
Homework Equations
I usually use the root test or with the limit of ##\frac {a_{n+1}}{a_n}##
The Attempt at a Solution
My first reaction is that this sum looks...
Homework Statement
I need to solve the DE
y’ = x^2y
using the power series method
Homework Equations
y = sum(0->inf)Cnx^n
y’ = sum(1->inf)nCnx^(n-1)
The Attempt at a Solution
I plug in the previous two equations into the DE. What is the general procedure for these problems after that...
I am reading Stephen Abbott's book: "Understanding Analysis" (Second Edition) ...
I am focused on Chapter 6: Sequences and Series of Functions ... and in particular on power series ...
I need some help to understand Theorem 6.5.1 ... specifically, some remarks that Abbott makes after the proof...
Homework Statement
Hello,
I need to find an expression for the sum of the given power series
The Attempt at a Solution
I think that one has to use a known Maclaurin series, for example the series of e^x. I know that I can rewrite
, which makes the expression even more similar to the...
Hello,
I was given f(-4x)= 1/(1+4x), and I used the geometric series to find the power series representation of this function. I then took the limit of (-4x)^k by using ratio test. The answer is abs. value of x. So -1/4<x<1/4
I then plugged in those end points to the series going from k=0 to...
Homework Statement
The Attempt at a Solution
I have deliberately made several obvious steps, because I keep ending up here. However I have no idea what to do from here. I thought about the fact, that differential equations have the solution ##x = x_{HOM} + x_{Inhom}##, but the ##x_{HOM}##...
Hi guys!
Here's a problem i was working on. I solved it by root test and got absolute value of x on the outside of the limit and the limit equaled zero. Is it wrong to multiply the outside absolute value by the zero I got from the limit? or is that okay?
In general, when we are solving power...
I've found the general solution to be y(x) = C1cos(x) + C2sin(x).
I've also found a recursion relation for the equation to be:
An+2 = -An / (n+2)(n+1)
I now need to show that this recursion relation is equivalent to the general solution. How do I go about doing this?
Any help would be...
Homework Statement
Suppose we are given the power series expansion ##f(x) = -\sum_{n=1}^{\infty} \frac{x^{n}}{n} ## which converges for |z|<1.
What is the radius of convergence?
Sum this serie and derive a power series expansion for the resulting function around -1/2, 1/2, 3/4 and 2.
The...
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...
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
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
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...
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
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...
Homework Statement
Homework Equations
Power series
ODE
The Attempt at a Solution
[/B]
Sorry for not typing all those things out from my phone..
How can I get C1?
And how can I put the solution in the required format? (I don't know how to put it in summation sign... and i cannot even solve...