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.
Homework Statement
I need to demonstrate that \frac{\mathrm{d} }{\mathrm{d} x}\sum_{n=0}^{\infty }\frac{x^{n}}{n!}= \sum_{n=0}^{\infty }\frac{x^{n}}{n!}
Homework EquationsThe Attempt at a Solution
I just need a hint on how to start this problem, so how would you guys start this problem?
Homework Statement
Homework Equations
We're using generating functions, and recurrence relations of homogeneous and non-homogeneous types
The mark allocation is 2, 3, 3 and 2
The Attempt at a Solution
I think I've done the first part correctly. The closed form is in terms...
Homework Statement
A heavy weight is suspended by a cable and pulled to one side by a force F. How much force is required to hold the weight in equilibrium at a given distance x to one side.
From classical mechanics, TcosX= W and TsinX=F. Find F/W as a power series of X(angle).
Often in a...
hi!
are the following power series equivalent?
ln(1+x)=\sum_{n=0}^{\infty} \frac{(-1)^n n! x^{n+1}}{(n+1)!}
=\sum_{n=0}^{\infty} \frac{(-1)^n x^{n+1}}{n+1}
Homework Statement
The problem is:
(x2 - 4) y′′ + 3xy′ + y = 0, y(0) = 4, y′(0) = 1
Homework Equations
Existence of power series:
y = \sum c(x-x0)^n
or
y = (x-x0)^r\sum c(x-x0)^n
The Attempt at a Solution
I know the point x=2 is an ordinary point of the differential...
Homework Statement
determine the radius of convergence of the given power series
\sum^{inf}_{n=1}\frac{n!x^n}{n^n}
Homework Equations
The Attempt at a Solution
I did the ratio test
then I had to take the 'ln'
but, my answer is this
|e|<1 for the series to converge.
It...
Homework Statement
Suppose that the power series \Sigma[a]_{n}[/tex]z^{n} and
\Sigma b_{n} z^{n} havr radii of convergence R! and R2 respictively. Prove that the radius
of convergence of the multiplication is at least R1 * R2
Homework Equations
The Attempt at a Solution
I...
Homework Statement
let an= \sum^{k=1}_{n} 1/\sqrt{k}
what is the radius of convergence of \Sigma\suma^{n=1}_{infinity} a_{n}x^n
i tired including the an term into the x^n equation then i got stuck.. help please
2. Suppose that \alpha and \beta are positive real numbers with...
Homework Statement
find the first three non zero terms of a power series representation of f(x)= sinh 2x
Homework Equations
The Attempt at a Solution
seems easy enough do I just substitute 2x for x?
so sinh 2x= 2x + 8x3/3! + 32x5/5!
Homework Statement
using the power series method (centered at t=0) y'+t^3y = 0 find the recurrence relation
Homework Equations
y= \sum a_{n}t^{n} from n=0 to infinity
y'= \sum na_{n}t^{n-1} from n=1 to infinity
The Attempt at a Solution
I went through and solved by putting the...
Homework Statement
Find the radius of convergence of
\sumn!*xn from n=0 to \infty
Homework Equations
The Attempt at a Solution
I did the ratio test and was able to get it down to abs(x) * lim as n approaches \infty of abs(n+1). It seems to me that the radius of convergence...
Im given the following power series:
\sum (x-3)^n/ n
I determined that the radius of convergence is R=1 and the interval of convergence is [2, 4)
They ask what values of x for which series converges absolutely?
and values of x for which series converges conditionally?
From what i...
Homework Statement
The firstthing I need to note is this is for a HISTORY of math course, so we have to use non modern techniques in most cases, some not. In other words, thequestion describes how to solve them. I'm also on a compyuter with a terrible keyboard so I'm doing my best.
1)...
Power Series ArcTan?
Homework Statement
Let f be the function given by f(t) = 4/(1+t^2) and G be the function given by G(x)= Integral from 0 to x of f(t)dt.
A) Find the first four nonzero terms and the general term for the power series expansion of f(t) about x=0.
B) Find the first four...
I just had this question on my exam and I was wondering whether my method of calculating it was right:
I was meant to find sum to n terms of this:
1, (4-a)x, (7-a^2)x^2, (10-a^3)x^3, (13-a^4)x^4...
Would the correct way to go about it be expand it:
1, 4x-ax. 7x^2-a^2x^2...
and...
Homework Statement
Use power series to estimate \int_0^1 \cos(x^2)dx with an error no greater than 0.005Homework Equations
Lagrange Error Formula \frac{f^{(n+1)}}{(n+1)!}(x-a)^{(n+1)}
The Attempt at a Solution
My original attempt was to find the series for \cos(x^2) , integrate it, and...
Homework Statement
evaluate ∑ n^2.x^n where 0<x<1
Homework Equations
The Attempt at a Solution
let a_n = n^2 and c=0
then radius of convergence, R=1
hence the series convergences when |x|<1
let f(x) = ∑ n^2.x^n
then f'(x) = ∑ n^3.x^n-1 for n=0 to infinity
then f'(x) = ∑...
Homework Statement
Consider the power series
Σanxn = 1+2x+3x2+x3+2x4+3x5+x6+…
in which the coefficients an=1,2,3,1,2,3,1,... are periodic of period p=3. Find the radius of convergence.
Homework Equations
The Attempt at a Solution
My attempt at a solution was to first state...
(Moderator's note: thread moved from "General Math")
Hi.
I am confused with this question. I tried two different ways to solve it, but I got different answers for each way. The question is
"Determine the coefficient of x^100 in the pwer series form of (1+x+x^2)/((1-x^3)^2)"
First, I tried...
Homework Statement
If A is a diagonal matrix with the diagonal entries a1, a2, ..., an, use the power series to prove that exp(At) is a diagonal matrix with the entries exp(a1t), exp(a2t), ..., exp(ant).
Homework Equations
The Attempt at a Solution
I can prove that A is...
Homework Statement
Let F be a field. Consider the ring R=F[[t]] of the formal power series
in t. It is clear that R is a commutative ring with unity.
the things in R are things of the form infiniteSUM{ a_n } = a_0 + a_1 t + a_2 t +...
b is a unit iff the constant term a_0 =/= 0...
Two Power Series questions that need to be solved urgently
Homework Statement
Question 1: The function f(x) = 2x (ln(1+x)) is represented as a power series. Find coefficients c2 through c6 of the power series.
Question 2: Write a partial sum for the power series which represents the function...
Homework Statement
Find the radius and interval of convergence of the power series:
\sum_{n=1}^{\infty} \frac{x^{2n-1}}{(n+1)\sqrt{n}}
Homework Equations
..
The Attempt at a Solution
My soltion:
the ratio test will gives |x^2|=|x|^2
it converges if |x|^2 < 1
i.e. if |x|<1
i.e...
Homework Statement
1.Is there any non-trivial power series that uniformly converges in all R? (a non-trivial power series has infinite non-zero coefficients...)
2. 2. Let f(X) be defined in [a,b]. We'll define fn(x) = [nf(x)] / n where [t]=floor value of t...
Check if the series (fn(x))...
Hi,
I am trying to prove something, but I need some kind of a result on the coefficients of a power series.
Suppose f(z) has a power series expansion about zero (converges). What can I say about the sum of the absolute values of the coefficients? Ideally I would like to show this sum is...
We have already shown 1+ w+ w^2 =0
If w is the complex number exp(2*Pi*i/3) , find the power series for;
exp(z) +exp(w*z) + exp (z*w^2)
We have already shown 1+ w+ w^2 =0
How would you go about finding the power series for sqrt(x+1) by applying the square root algorithm. I can do it using binomial expansion and other formulas but I'm not familiar with the square root algorithm involving variables.
[b]1. Find a power series for F(x)= 3/4x^3-5, where c=1
[b]2. power series = 1/a-r
[b]3. What I did was take a derivative to get a similar function that was easier to solve. I used 1/x^3-1. Then I found a series for that function. Which I got \sum(x^3)^n. Then I added back from my...
Homework Statement
Find a third degree polynomial approximation for the general solution to the differential equation:
\frac{d^{2}y}{dt^{2}} +3\frac{dy}{dt}+2y= ln(t+1)
Homework Equations
Power series expansion for ln(t+1)
The Attempt at a Solution
The system to the...
Using the power series method to solve the differential equation
y'+xy=0 when y(0)=1
Write the solution in the form of a power series and then recognize what function it represents.
************************************
My answer:
\sum(-1)k*[(x2k)/(2k)*k!]
Is my answer correct?
Is...
Homework Statement
y''+t^2*y'-y=1-t^2Homework Equations
y(0)=-2
y'(0)=1
Find the first 6 coefficients
C1=-2
C2=1
C3=? (-1?)
C4=?
C5=?
C6=?
The Attempt at a Solution
Okay so I tried to do this but I'm not used to having anything on the RHS of the equation.
I got down to...
Homework Statement
"Find the radius of convergence of the power series for the following functions, expanded about the indicated point.
1 / (z - 1), about z = i.
Homework Equations
1 / (1 - z) = 1 + z + z^2 + z^3 + z^4 + ... +
Ratio Test: limsup sqrt(an)^k)^1/k
The...
I am having trouble getting to a solution for this differential equation
2(x^2+2x)y' - y(x+1) = x^2+2x -------- 1
for a series solution, we have to assume y = \sum a_{n}x^n ---------- 2
if we divide equation 1 by x^2 + 2x , we get (x+1)/(x^2+2x) for the y term, which is where my problem...
Hey guys. I'm new here. I've been trying to figure out how to solve this problem, and I'm still confused.
(-x^2 + 4x -3)* d2y/dx2 - 2(x-2) * dy/dx + 6y = 0
y(-2) = 1
dy/dx(-2) = 0
I set y = \suman(x+2)n (start at n=0, n goes to infinity)
dy/dx = \sumann(x+2)n-1 (start at n=0, n...
given the infinite power series
f(x)= \sum_{n=0}^{\infty}a_ {n}x^{n}
if we know ALL the a(n) is there a straight formula to get the coefficients of the b(n)
\frac{1}{f(x)}= \sum_{n=0}^{\infty}b_ {n}x^{n}
for example from the chain rule for 1/x and f(x) could be obtain some...
Homework Statement
Say f(z) = Σ(z^n), with sum from 0 to infinity
Then we can say f'(z) = Σn(z^n-1), with sum from 0 to infinity (i)
= Σn(z^n-1), with sum from 1 to infinity (as the zero-th term is 0)...
Homework Statement
Find tan(z) up to the z^7 term, where tan(z) = sin(z)/cos(z)
Homework Equations
sin(z) = z - z^3/3! + z^5/5! - z^7/7! + ...
cos(z) = 1 - z^2/2! + z^4/4! - z^6/6! + ...
The Attempt at a Solution
Hi,
Seeing as sin and cos have the same power series as for when...
Homework Statement
Find the power series for the function
f(z) = (1-z)^-m
Hint: Differentiation gives:
f'(z) = m(1-z)^m-1
= m(1-z)^-1.f(z)
or:
zf'(z) + mf(z) = f'(z)
Use the formula for differentiation of power series to determine the coefficients of the power series for f...
Homework Statement
Consider the matrix [1,-5,5;-3,-1,3;1,-2,2]
Do four interations of the power method, beginning at [1,1,1] to approximate the dominant eigenvalues of A
Homework Equations
Matrix multiplication
The Attempt at a Solution
Okay my issue with this problem is this
I...
Homework Statement
using the fact that \frac{sint}{t} = \sum^{}_{n=0}\frac{ (-1)^{n} t^{2n}}{(2n+1)!}
i wanted to write:
using that sint/t = ∑ (-1)ⁿ·t²ⁿ/(2n+1)!
find a power series solution of the equation tx'' +sint x = 0
Homework Equations
sorry i pushed the wrong button,The Attempt at a...