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
(a) Expand f(x) as a power series
f(x)=\frac{7}_{\sqrt[4]{1+\frac{x}_{14}}}
Which I converted to...
7 - \frac{1}{8}*x + \sum^{\infty}_{n=2}7*(-1)^{n}*\frac{1*5*9*...*(4n-3)}{4^{n}*n!}*(\frac{x}{14})^{n}
(b) Use part (a) to estimate 7 / (1.1)^(1/4) correct to...
By Weierstrass approximation theorem, it seems to be obvious that every continuous function has a power expansion on a closed interval, but I'm not 100% sure about this. Is this genuinely true or there're some counterexamples?
I'm in a "series and sequences" class. Up until power series, if I studied the material a decent amount and did a lot of practice problems, things made sense.
We got to power series today and I simply am not getting anything. I read the chapter in the book just now and it totally lost me...
Homework Statement
Find the radius of convergence and interval of convergence of the following series.
\sum^{\infty}_{n=1}\frac{n!*x^{n}}_{5*11*17*\cdots*(6n-1)}
Homework Equations
Knowledge of Power Series.
Factorials?
The Attempt at a Solution
I'm really unsure where to start...
Homework Statement
Evaluate the integral of xarctan(3x) from 0 to 0.1 by expressing the integral in terms of a power series.
Homework Equations
The Attempt at a Solution
I differentiated xarctan(3x) until I got two functions that I could turn into power series (arctan(3x) and...
Homework Statement
Solve the initial value problem y'' = y' + y where y(0) = 0 and y(1) = 1
derive the power series solution y(x) =
\ \ \sum_{n=1}^{\infty}{(F_{n}x^n)/n!} \ \ where {Fn} is the sequence 0,1,1,2,3,5,8,13... of Fibonacci numbers defined by F0 = 0 and F1 = 1
Homework...
Homework Statement
Solve this equation using power series: y'' + y = x
Homework Equations
none
The Attempt at a Solution
I am confused about the x on the RHS of the equation. If the equation was y'' + y = 0, I would have no problem solving it. I am just a little confused...
Homework Statement
There is a power series
\infty
\sumb_k.z^k
n=0
such that
\infty
(exp(z) - 1)\sumb_k.z^k = z
n=0
the infinity and n=0 are meant to be over the sigma, sorry
Find b_k for k...
Homework Statement
Find the power-series expansion about the given point for the function; find the largest disc in which the series is valid.
f(z) = z^3 + 6z^2-4z-3 about z0=1.
Homework Equations
The Attempt at a Solution
The series is fine. Since it's a polynomial, there are only three...
Homework Statement
Find a power series representation for the given function using termwise integration.
f(x) = \int_{0}^{x} \frac{1-e^{-t^2}}{t^2} dt
Homework Equations
The Attempt at a Solution
Well, I figured I could rewrite it like this using the Maclaurin series for...
Homework Statement
Compute \sum_{n=0}^{\infty} p^n cos(3nx) for \abs{p} \textless 1 , where p \in \mathbb{R} .
Homework Equations
The Attempt at a Solution
I was thinking that maybe this could be approached as a telescoping series, but I'm not really sure if it is. Would that...
I was going through http://mathworld.wolfram.com/LaguerreDifferentialEquation.html" in Wolfram which gives brief details about finding a power series solution of the Laguerre Differential Equation. I was reading the special case when v = 0.
I read earlier from Differential Equations by Lomen...
Homework Statement
Find a power series representation for f(x) using termwise integration, where f(x) = \int_{0}^{x} sin(t^3) dt .
Homework Equations
The Attempt at a Solution
I've never done this before, but apparently, if I have a power series representation for sin(t^3), I...
Homework Statement
Find the radius of convergence of the series:
∞
∑ n^-1.z^n
n=1
Use the following lemma:
∞ ∞
If |z_1 - w| < |z_2 - w| and if ∑a_n.(z_2 - w)^n converges, then ∑a_n.(z_1 - w)^n also...
Homework Statement
The function
f(x)=\frac{1}{1+x^{9}}
can be expanded in a power series
\sum^{\infty}_{0} a_{n}x^{n}
with center c = 0.
Find the coefficient
a_{27}
of
x^{27}
in this power series.
2. The attempt at a solution
I can get to:
\sum^{\infty}_{0} (-1)^{n}(-x^{9})^{n}
which I...
Homework Statement
I am trying to figure out how to represent an inverse laplace transform by a power series. There is an example in my book but it is not very well explained.
f(s)=1/s+1 which i know is the transform of y=e^-t.
In the book they use the fact that L(t^n)= n!/s^n+1. and...
Homework Statement
Give an example of a power series tha converges on the interval (3,6), but on no larger interval. Give some justification.
Homework Equations
The equation needed is probably that for a power series:
\Sigma cn(x-a)n
The Attempt at a Solution
I'm not sure at all. The...
Homework Statement
Suppose the series \sum_{n=0}^{\infty} a_n x^n has radius of convergence R and converges at x = R. Prove that \lim_{x \to R^{-}}\large( \sum_{n = 0}^{\infty} a_n x^n \large) = \sum_{n = 0}^{\infty} \large( \lim_{x \to R^{-}} a_n x^n \large)
2. Question
For the case R...
Homework Statement
If a_v > 0 and \sum_{v=0}^{\infty} a_v converges, then prove that \lim_{x \rightarrow 1^-} \sum_{v=0}^{\infty} a_v x^v = \sum_{v=0}^{\infty} a_v .
Homework Equations
The Attempt at a Solution
Since \sum a_v converges, then we can say that \sum a_v x^v converges for x...
Find a power series sol'n: (x2-1)y'' + 3xy' + xy = 0
Homework Equations
let y = \Sigma (from \infty to n=0) Cnxn
let y' = \Sigma (from \infty to n=1) nCnxn-1
let y'' = \Sigma (from \infty to n=2) n(n-1)Cnxn-2
The Attempt at a Solution
I wrote the differential eq as...
Hi,
I have 2 questions regarding how to expand power series.
1). Find the power series expansion of Log z about the point z= i - 2
2). Expand the function 1/(z^2 + 1) in power series about infinity
Any help will be greatly appreciated. This is because I am totally unsure about what...
Please can somebody help me with this problem
y" + y' + sin^2(x)y - 2sinx = 0
I used power series method and i used the macclurin expresion for sinx but i was not able to get a recurrence formula.
I need to find \frac{1}{i+z} as a power series in z.
I want to know if am doing this right.
If i use the taylor series here by doing
f(z) = z^i
f'(z) = i z^{-1} z^i
f''(z) = i (i-1) z^{-2} z^i
This taylor series is just for z= i...
Homework Statement
I'm having a little trouble concerning the part where we have to calculate the endpoints of a known interval of convergence in order to see whether they are convergent or divergent. In this case, is the summation of n=0 to infinity of (-1)^n/10 diverging or converging. Are...
Homework Statement
I am confused about how to find a sum of a power series, especially when it contains factorials and I can't quite get it to look like a geometric series. Is it the same thing as finding a limit (and then I would follow the various tests for convergence of the different...
Hello, I need to find the sum(as a function of x) of the power series \Sigma^{\infty}_{n=0}\frac{(x+1)^n}{(n+2)!}
The hint i was given was compare it to the Taylor series expansion of ex.
Im not sure even how to start this problem and any help is much appreciated.
How to do expansion as power series of any random function??
The template doesn't really apply because this is a general question rather than a specific problem.
If I am given some function, how do I expand it as a power series? For the past two semesters of my physics degree, I have been...
Homework Statement
Hi all.
In my book on complex analysis, they discuss complex power series. They use a variety of "tests" to determine absolute convergence, but they never say if this also implies convergence.
Does it?
Niles.
Which of the following expansions is impossible?
a \sqrt{x-1} in powers of x
b \sqrt{x+1} in powers of x
c ln(x) in powers of (x-1)
d tanx in powers of (x-\pi / 4 )
e ln(1-x) in powers of x
What are htey asking and how do i do this? the answer is A by the way
Homework Statement
How do I prove a power series is onto? Since I cannot calculate directly, especially I haven't learned Jodarn Normal form.
Homework Equations
The Attempt at a Solution
By showing 1-1, I tried
∑(1/n!)[(M)^n-(N)^n]=0, what can I conclude from this step?
Homework Statement
In the following series':
http://image.cramster.com/answer-board/image/cramster-equation-2009410014306337491927047975008434.gif
According to my book, we only have a common range of summation here for n >= 2.
Therefore we need to treat n = 0 and n = 1 separately...
Homework Statement
Find the radius and interval of convergence of the given power series.
\sum^{\infty}_{n=0}\stackrel{100^{n}(x+7)^{n}}{n!}
Homework Equations
Ratio Test
The Attempt at a Solution
My real question is: Can the radius be 0? Cuz that's what I get. Would the...
Homework Statement
The problem just states to find the Laplace Transform of cos(kt) from its power series expansion, instead of using the formula for the transform of a periodic function.Homework Equations
Equation for Laplace transform of a function f(t) ->\int(e^{-st}f(t))dt
Power Series...
Homework Statement
Suppose the real power series \sum ^{\infty}_{n=0}c_{n}x^{n} has radius of convergence R > 0. Define f:= \sum ^{\infty}_{n=0}c_{n}x^{n} on I:= (-R, R) and let b \in I. Show that there exists a power series \sum d_{n}(x-b)^{n} that converges to f(x) for |x-b| < r - |b|...
When solving diff-eq's given initial values, e.g.
y'' - 2y ' + y = 0
y (0) = 0
y ' (0) = 1
Can one assume immediately that
y(0) = c0
and y ' (0) = c1
?
Since these are the first 2 terms in the series?
Thanks!
Homework Statement
1. Find a power series for (1-x)^(-1/2)
2. Find a power series for (1-x^2)^(-1/2)
3. Find a power series for arcsin(x)
Homework Equations
Binomial series, (1+x)^k= 1 + kx + k(k-1)x^2/2!+ k(k-1)(k-2)x^3/3!+...
The Attempt at a Solution
for 1., I have 1+ (-1/2)(-x) +...
Homework Statement
Determine the series of the given function:
f(x) = 10 / (1-5*x)
Homework Equations
Power series of 1/(1-x) = Σ from n=0 to n=infinity of (x^n) The Attempt at a Solution
f(x) = 10/(1-5x)
= 10*(1/1-5x)
= 10 * Σ(5x)^n
= 10 * Σ(5^n)*(x^n)
= Σ (50^n)*(x^n) <--- Not sure if...
Homework Statement
Word for word, the book says "Find a power series that represents 1/(1+x^2) on (-1, 1)
Homework Equations
It's in the chapter that talks about power series, so I think they want me to use the fact that 1/(1-x) is a power series with a=1 and r=x, but if I just...
Hi everyone :smile:
When determining the radius of convergence of a power series, when should I use the ratio (a[sub n+1] / a[sub n]) test versus the root (|a[sub n]|^(1/n)) test?
I know that I'm supposed to use the ratio only when there are factorials, but other than that, are these tests...
Homework Statement
Solve
(1-4x^2)y''+34x\cdot y'-70y=0
Homework Equations
Basically, I found the recurrence relationship to be:
a_{n+2}=\frac{2 (-7 + n) (-5 + 2 n)}{(n+1)(n+2)}a_n}
Now, I solve for y1 where y1 had a_0=0 and a_1 = 1. It is a simple polynomial of degree 7...
How do I go about finishing/calculating this?
Homework Statement
Use a power series to approximate \int\cos 4x\log x dx to six decimal places. (bounds are from pi to 2pi)
Homework Equations
The Attempt at a Solution
So I broke down the equation first:
\int\cos 4x\log x dx =...
Hey there! I'm new here and I just want to ask anyone willing how to solve this problem using power series:
y"+3y'+2y= sin x
y(0)=0
y'(0)=1
Evaluate y(0.1)
Thanks! :smile:
Homework Statement
[Directions to problem]
Show that the function of x gives a power series expansion on some interval centered at the origin. Find the expansion and give its interval of validity.
\int_0^x e^{-t^2} dt
Homework Equations
The Attempt at a Solution
I have...
Homework Statement
power series expansion of:
((cosh x)/(sinh x)) - (1/x)
Homework Equations
cosh x = (1/2)(ex + e-x)
sinh x = (1/2)(ex - e-x)
The Attempt at a Solution
what i have so far:
I simplified the first part of the eq to read :
e2x-1
e2x-1
now I am stuck...
Homework Statement
Find the closed form of the following power series
1+3x+6x^2+10x^3+15x^4+21x^5+...
Homework Equations
1+x+x^2+.. = 1/(1-x)
The Attempt at a Solution
I tried differentiating but couldn't get it to any expression that I know the sum for.. I was playing around...
I have to find the "second smallest root" of the following equation :
1-x+(x^2)/(2!)^2-(x^3)/(3!)^2+(x^4)/(4!)^2+...=0
Matlab returns quite a satisfactory answer. >> p=[1/518400 -1/14400 1/1576 -1/36 1/4 -1 1]
p =
0.0000 -0.0001 0.0006 -0.0278 0.2500 -1.0000 1.0000...
Homework Statement
\begin{equation}
1 - x + \frac{x^2}{(2!)^2} - \frac{x^3}{(3!)^2} + \frac{x^4}{(4!)^2} +... = 0 \nonumber
\end{equation}
Homework Equations
To find out the power series in the LHS of the given equation.
The Attempt at a Solution
I have tried to solve it by...
I have to find the power series f(x)= 3 / (2 -5x).
I devided everything by 2 so I have (1/2) / (1 - 5x/2) = \sum ar^n
and then through some steps I have 15x^n / 2^n+1
and I on the right track? How do I find the interval of convergence?