Questions tagged [power-series]
Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n (x-c)^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex.
1
vote
1answer
22 views
Calculate coefficient $z^n u^j$ of power series in two variables
I'm trying to calculate
$$[z^n u^j] \frac{1}{(1-zu)(1-z)} \log \left(\frac{1}{1-zu}\right),$$
where $[z^n u^j] \sum_{n=0}^\infty \sum_{j=0}^\infty F_{n,j} z^n u^j = F_{n,j}$.
So I have to ...
-2
votes
0answers
11 views
Equation to get xi based of sum of xn [on hold]
I want to write code/equation that return Xi value base on this logic.
Assuming X a positive non decimal number >1, and n ...
1
vote
1answer
31 views
How does this method work to find prime numbers?
I'm curious about this pattern that I saw while adding many powers of two together, and then taking the prime factorisation of each result, and I'm curious as to why this occurs.
Here is the pattern:
...
4
votes
2answers
83 views
Find the zeros of $f(z)=z^3-\sin^3z$
I want to find the zeros of $f(z)$, $$f(z)=z^3-\sin^3z$$
My attempt
$f(z)=0$
$z^3-(z-z^3/3!+z^5/5!-\dots)^3=0$
$z^3-z^3(1-z^2/3!+z^4/5!-\dots)^3=0$
$z^3[1-(1-z^2/3!+z^4/5!-\dots)^3]=0$
So $z=0$ ...
0
votes
0answers
43 views
Uniform convergence $\sum_{n=1}^\infty \frac{n^2-n^4}{n^5+n^3+1}\left(x^2\sin\left(\frac{\pi x}{2}\right)\right)^n$. Troubling finding the interval.
I have to find the interval of uniform convergence, for the series
$$\sum_{n=1}^\infty \frac{n^2-n^4}{n^5+n^3+1}\left(x^2\sin\left(\frac{\pi x}{2}\right)\right)^n$$
Now my attempt has been to use ...
3
votes
1answer
29 views
Closed-form expression for infinite series related to a Gaussian
Consider the following infinite series, where $x$ is indeterminate and $r$ is held constant:
$\displaystyle 1 + \frac{x}{r} + \frac{x^2}{r^2} + \frac{x^3}{r^3} + ...$
It is relatively easy to see ...
2
votes
0answers
32 views
Uniqueness of the derivative of a function
Let $U\subset\mathbb{C}$ be an open subset of the complex plane.
Consider $C^\omega(U)$, the space of all functions $f:U\rightarrow \mathbb{C}$ such that $f$ has a convergent power series ...
2
votes
0answers
28 views
Is a power series with integral coefficients with a limit at x=1 a polynomial?
If we have a power series with integral coefficients $f(x)=\sum_{i=0}^{\infty}a_ix^i$ which converges for all $|x|<1$, and the limit as $x\rightarrow 1$ exists, where we approach along the real ...
1
vote
0answers
23 views
Find the radius of convergence R for power series $\sum_{i=0}^{\infty} n^a z^n$
For power series $\sum_{i=0}^{\infty} n^a z^n$
The series seems to be diverging as $C>1$. How can I find the radius of convergence R at this instance?
1
vote
1answer
26 views
measure the speed of convergence of two functions
Let's say I have two functions $f$ and $g$ such that :
$$\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = l$$
Then I can say that :
$$f \sim_a g$$
So when I say that two functions are equivalent near ...
1
vote
0answers
29 views
Verify this Proof: $\sum_{k=0}^{n}{n\choose k}=2^n$ via Taylor series
I've found this proof which I am quite proud of. Am I missing anything?
Theorem: $$\sum_{k=0}^{n}{n\choose k}=2^n$$
Proof:
Let $\beta\in\{t\in\Bbb R:t>0\}$, and $f(x)=x^\beta$ be continuous on ...
0
votes
0answers
31 views
Limit of a complex power series
let $f(z) = \sum_{n = 0}^{\infty}z^{n!}$
How can I calculate, given $ \omega \in S^{1}, the \lim_{z \to \omega} f(z)$
I understand that, like the geometric series, the limit may exists even if the ...
3
votes
3answers
53 views
re-calculating exponential sum when exponent changes
If $A_c$ can be calculated as follows:
$A_c=\sum_{k=1}^{N} a_ke^{ck}$
Where c is a known real constant and $a_k$ is a known series comprising real numbers which cannot be described by a function $f(...
0
votes
2answers
93 views
Sum of the First n Natural Numbers Power n
How would I estimate the sum of a series of numbers like this: $$1^n+2^n+\cdots+n^n$$
2
votes
3answers
56 views
$\sum\limits_{n=1}^{\infty} \frac{1}{ne^{n}}$ converges
Here, The WolframAlpha calculates this series :
$\sum\limits_{n=1}^{\infty} \frac{1}{ne^{n}} = 1 - \ln (e -1)$
But they do not explain how they do this calculation. Does anyone know how to do it?
...
0
votes
1answer
38 views
Conjecture: $(x^\alpha+p)^\beta=\,_1F_0(\beta;;x^\alpha+p-1)$ and ideas for proof
Conjecture: For $|z|<1,\,\alpha,\beta,p\in\Bbb R$
$$(z^\alpha+p)^\beta=\,_1F_0(\beta;;z^\alpha+p-1)$$
I found this formula by noting that $$z^\alpha=\,_1F_0(\alpha;;z-1)$$
Via the simplification of ...
1
vote
1answer
34 views
Power series radius of convergence and if they are divergent or not
$\sum_{i=0}^{\infty} e^{-\sqrt n}z^n $
I tried to find the radius of convergence of a power series.. is this equation a geometric series?
or would it be easier to do a ratio test and
$\lim_{n\to\...
1
vote
0answers
25 views
Lifting finite order automorphism of power series ring
$\require{AMScd}$
Let $\mathcal{O}$ be an (finite) extension of the $p$-adic integers.. Denote by $R$ the power series ring $\mathcal{O} [[X_1, \dots, X_n]]$ and let $I$ be an ideal such that $R/I$ ...
22
votes
4answers
2k views
Why doesn't the Stone-Weierstrass theorem imply that every function has a power series expansion?
I know that not every function has a power series expansion.
Yet what I don't understand is that for every $C^{\infty}$ functions there is a sequence of polynomial $(P_n)$ such that $P_n$ converges ...
1
vote
1answer
28 views
convergence radius of power series with $x^{n^2}$
can someone please help me understand how can i find convergence radius for the following power series:
$$
\sum_{n=1}^\infty \frac{x^{n^2}}{2^{n-1} n^n}
$$
I tried positioning $t=x^n$ and i found ...
-1
votes
0answers
19 views
Write power series in closed form
I have no idea how to begin to try and write this in closed form... $$p(x)=\sum_{i=0}^\infty (3+2x)^i $$
I know doing $i=1, 2, 3$ is equal to $1+(3+2x)+(9+12x+4x^2)$
0
votes
0answers
26 views
Extreme-mega-ultra-crazy hypergeometric functions
I've been lusting over hypergeometric functions, and came up with some questions. Here goes. I've defined the following functions, and I want to know if there are any closed forms for them.
$$f_1(2;...
0
votes
1answer
27 views
Solve second order differential equation with cosh using frobenios method
i need to show that the differential equation
$y^{''}+(\cosh(2x)-4)y = 0$
has the solution:
$ y(x) = x+\frac{1}{2}x^3-\frac{1}{40}x^5 -... $
using Frobenius method.
I started by writing cosh(2x) ...
0
votes
0answers
30 views
Clarification for Calculus Question (I don't need an answer, just interpretation)
The question is as follows:
Write out the first 5 terms of the power series.
$\sum_{n=0}^\infty \frac{3^n}{n!}x^{3n+1}$
This is a web-based system and all I get ...
0
votes
1answer
17 views
Find the radius of convergence for this power series
So originally I needed to turn the function $f(x)=\frac{3}{2-x}$ into a power series. I think I did this successfully and got $$\sum_{n=0}^\infty \frac{3x^{n}}{2^{n+1}} $$ Now I'm struggling to find ...
2
votes
1answer
26 views
Bürmann-Lagrange formula for a function
How to prove that, if
$$
w=ze^{aw^m} \quad (m\in\mathbb{N}),
$$
then
$$
w=\sum_{n=0}^\infty\frac{a^n(nm+1)^{n-1}} {n!}z^{nm+1} \; ?
$$
I am aware of the Bürmann-Lagrange inversion formula, that if $z=...
0
votes
0answers
32 views
Help working out the following series on the euclidean grid.
I am trying to approximate the following series on the euclidean grid where $\sqrt{i^2 + j^2}$ in the exponent comes from distances on the euclidean grid from the origin.
$x = \sum_{i,j \geq 0} e^{-a ...
-1
votes
2answers
47 views
First three coefficients of $f(x)=\sum_{n=0}^{\infty}\frac{1}{\sqrt{1+n}}x^n$
Given $f(x)=\sum_{n=0}^{\infty}\frac{1}{\sqrt{1+n}}x^n$ find the first three coefficients of the power series $\frac{1}{f(x)}=\sum_{n=0}^{\infty} b_n x^n$.
I tried the following: rewrote to $(1+n)^\...
-3
votes
2answers
55 views
Solve $\sum_{k=0}^\infty\frac1{k!}$ without Taylor expansion or any expansion.
Evaluate $\sum_{k=0}^\infty\frac1{k!}$ without Taylor, Mclauren or any expansion..
So I know that $$f(x)=e^x=\sum_{k=0}^\infty\frac{x^k}{k!}$$
around $0$ and for $x=1$ we have that: $\sum_{k=0}^\...
4
votes
0answers
96 views
On the series expansion of $\frac{\operatorname{Li}_3(-x)}{1+x}$ and its usage
Recently I asked about the evaluation of an integral involving a Trilogarithmic Function (which can be found here). Pisco provided a quite elegant approach starting with a recurrence relation of the ...
-2
votes
0answers
28 views
Sum of a power series with square root and fraction [closed]
Consider the power series
$$\sum_{n=0}^\infty \frac{x^n}{\sqrt{n+1}}$$
Find an approximate value for the sum for $x = −1$, within a tolerance of $0.1$
I know how to sum a basic power series. The ...
0
votes
2answers
30 views
Definite integral with power series
I need to approximate the following integral with six exact decimals using power series:
$$\int_{0}^{0.2} dx/(1+x^5) $$
What I have so far :
$$\int_{0}^{0.2} dx/(1+x^5) = (0.2)-(0.2)^6/6+(0.2)^{...
0
votes
0answers
35 views
Infinite sum $S(k) = \lambda + 2^k \lambda^2 + 3^k \lambda^3 + \dots$
I'm trying to find an expression for:
$$S(k) = \sum_{n=0}^{\infty} n^k \lambda^n$$
I have found a recursive expression for this where I first find $S(0)$, then use that to find $S(1)$. Then use those ...
7
votes
1answer
81 views
Closed form sum for the following series on the euclidean grid.
I am trying to find a closed form solution for the following series. The $\sqrt{i^2 + j^2}$ in the exponent comes from distances on the euclidean grid from the origin.
$x = \sum_{i,j} e^{-\sqrt{i^2 + ...
0
votes
3answers
31 views
Summation of a geometric sequence from $1$ to infinity for: $(n^{2})\times ((\frac{5}{6})^{n-1})$
I'm fully comfortable with most series and even arithmetico–geometric sequence including n to any exponent if the geometric term is in the form of $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$, and so forth....
0
votes
1answer
30 views
Find the sum explicitly in the interval of convergence of the series
Find the sum explicitly in the interval of convergence of the series.
$\sum _{n=1}^{\infty }\:\frac{n}{n+1}\left(2x-1\right)^n$
Here is the work that I did:
$let\:t\:=\:2x-1$
$\frac{n}{n+1}\:=\:1\:...
2
votes
2answers
24 views
Finding series of two functions multiplied with each other
Let's say we are given a function defined as ${\frac{\ln(1+t)}{1-t}}$. We want to find the series expansion up to ${t^4}$.
Now we know that we have two function within this larger function which are ...
0
votes
1answer
25 views
Radius convergence of ODE power series solution
I have to solve the ODE
$$y''-5xy'+y=0$$
using power series.
I found
$$y(x)=a_{0}\left(1-\frac{1}{2!}x^{2}-\sum_{k=2}^{\infty}\frac{9\cdot19\cdot\ldots\cdot(10k-11)}{(2k)!}x^{2k}\right)+a_{1}\left(x+...
1
vote
1answer
41 views
Power series expansion for $\sin(2x)\cos(x)$
So my objective is to find the power series expansion of
$$\sin(2x)\cos(x)$$
This is what I know so far, I just need a little help find the series:
First of all, we know that ${\sin(2x)=\sum\...
1
vote
1answer
45 views
Laurent expansion for $1/\cos(z)$
I have a quick question.
How to find the Laurent expansion for $1/\cos(z)$
In the link above the person asks how to find the Laurent expansion for $\frac{1}{cos(z)}$.
The accepted answer utilizes ...
0
votes
0answers
12 views
Convergence radius estimator with three consecutive terms
The well known ratio estimator for the convergence radius of a Taylor series is $\left| \frac{c_k}{c_{k+1}}\right|$ and it uses two consecutive terms in the Taylor series, but does not always work. E....
0
votes
1answer
13 views
Z transform and non-$\ell^{1}$ causal sequences
Consider a complex sequence $h[n]$, such that:
if $n < 0 \Rightarrow h[n] = 0 $
$\sum_{n=0}^{+\infty} \left|h[n]\right|$ diverges
$H(z) = \sum_{n=0}^{+\infty} h[n]z^{-n}$ converges uniformly $\...
1
vote
1answer
38 views
Power Series / Taylor Expansion of $\frac{1}{\left(1-t^2\right)^{\frac{1}{2}}}$
The question:
$\left(\frac{1+t}{1-t}\right)^{\frac{1}{2}}\:=\:\sum _{n=0}^{\infty }\:a_nt^n$
The question asks for what is $a_n$
These are the steps I've done:
$\left(\frac{1+t}{1-t}\right)^{\frac{...
0
votes
1answer
21 views
If a power series converges uniformly to f on some open U and f in C1 on R, does the series converge everywhere?
Now let me be more precise with my question:
If $\sum c_kx^k = f(x)$ for $x\in U$, where $U$ is some open disc, and if furthermore $\sum c_kx^k$ converges uniformly on $U$ and $f(x)\in C^1(R)$, then ...
-1
votes
0answers
16 views
Finding ratio of convergence of complex valued power series.
I am trying to find the radius of convergence of the following series:
$ \sum_{k=2}^{\infty} k^{log(k)}(z+1)^k$
I have tried the ratio and root test but they both seem to go no where? I can not see ...
3
votes
0answers
61 views
Replacing $n!$ with Stirling's approximation in $e^x = \sum_n \frac{x^n}{n!}$
I was wondering if there is a closed-form expression for
$$\sum_{n=0}^{\infty} \frac{x^n}{e^{-n}n^n},$$
although I expect there is none because Mathematica cannot compute it. However, from Stirling'...
1
vote
2answers
46 views
Study simple convergence of $\sum_{n=0}^{\infty} x^{2n}$ on $[0,1[$
I have to study
1 ) the simple convergence of
$$S(x) = \sum_{n=0}^{\infty} x^{2n}$$
and
2) the uniform convergence
My attempts :
1)
$\forall x \in [0,1[$ $$S_n(x) = \frac{1-(x^2)^{n+1}}{1-x^2}...
2
votes
2answers
58 views
Find the radius of convergene R for power series
For power series, find the radius of convergence R and determine if it is conditionally convergent, absolutely convergent, or divergent for $z = R$ and $z = −R$.
$\sum_{i=0}^{\infty} e^n z^n$
I'm ...
0
votes
1answer
53 views
Prove endpoints of Convergence Interval of Power Series are divergent
Part a) of a question required showing that the radius of convergence of the power series $\sum_{} \frac{n!}{n^n}x^n$ and $\sum_{} \frac{n^n}{n!}x^n$ are e and 1/e respectively. This was fairly simple....
0
votes
1answer
15 views
Show $\exists !\mathbf{c} = \mathbf{0}$ when $\mathbf{c}^Tf(\mathbf{x}) \equiv 0$, where $f$ is a vector function consisting of polynomials of $x_i$'s
For $\mathbf{x} = (x_1, x_2, \cdots, x_\nu) \in \mathbb{R}^\nu$, where $\nu = \max \{ j(1), \cdots, j(n) \} $, define $f :\mathbb{R}^\nu \rightarrow \mathbb{R}^n$ as follows.
$$
f(\mathbf{x} ) = \...
