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.
7,607
questions
0
votes
1answer
25 views
Calculating inverse of power series
I found this calculation for the inverse of power series: $$x^2-4x+4.$$
This is the solution:
$$
\frac{1}{x^2-4x+4} =
$$
$$
\left(\frac{1}{2-x}\right)^2 =
$$
$$
\frac{1}{4} * \left(\frac{1}{1-\frac{...
0
votes
0answers
24 views
Taylor expansion in one variable of two variable function
Can I write the expansion of $e^{xy}$ as
$$e^{xy}=\sum_{n=0}^{\infty}\frac{y^n}{n!}f_n(x)$$
Where
$$f_n(x)=\frac{\partial^n}{\partial y^n}e^{xy}|_{y=0}$$
-1
votes
0answers
11 views
Finding singular points on x axis [closed]
locate and classify singular points on the x-axis for following differtial equation
(x^2)y''+(2-x)y'=0
-2
votes
0answers
32 views
What is power series solution of xy'=y [closed]
What is the Power series solution of xy'=y.
I ended up with all coefficients zero
3
votes
3answers
85 views
$\lim_{n \to \infty}\frac{\sum_{i=0}^{n}{\sqrt{i}}}{n^2}=0 \; ?$
I was wondering whether this limit converges to zero:
$$
\lim_{n \to \infty}\frac{\sum_{i=0}^{n}{\sqrt{i}}}{n^2}=0
$$
And i'm pretty sure it is.
First, by intuition. I know that $\sum_{i=0}^n{i} = \...
0
votes
0answers
40 views
Discrete power series?
I have a pattern. I have a box made of $n\times n\times n$ boxes where n is a power of 2.
I flatten this box into an $n^3$ flat sequence of boxes.
I then have an $n/2 \times n / 2 \times n/2$ box, ...
0
votes
0answers
27 views
What is the radius of convergence of $f(x)=\int_{0}^{\infty}t^{x^2-1}e^{-2t}dt$ written as a power series in a neighborhood of $0$?
Let $\displaystyle f(x)=\int_{0}^{\infty}t^{x^2-1}e^{-2t}dt$
Can $f(x)$ be written as a power series in a neighborhood of $0$?
Note that changing variables we can write $f(x)=2^{-x^2}\Gamma(x^2)$ ...
0
votes
0answers
26 views
A question about analystic function $ f$defined in a Angular domain.We want to show $f =0$ by adding some extra condition.$
Problem:let$G=\{z:0＜argz＜{π\over4}\}$. $f$ is analytic on $G$ while is continuous on $\bar G$. If $f=0$ on interval $[a,b]$ lining in the real axis,show that f equals 0 on G.
My attempt:I never see ...
3
votes
3answers
69 views
Faster way to find the first four non-zero terms of the Maclaurin series for $\frac{1-x}{1+x}\cosh\sqrt{x}$
I want to find the first 4 non-zero terms for :
$$\frac{1-x}{1+x}\cosh\sqrt{x}$$
Before expanding, I rewrite this as $$(1-x)\left(\frac{1}{1+x}\right)\cosh\sqrt{x}$$
Then I expand to get $$(1-x)\left(...
0
votes
0answers
39 views
Prove using the power series representation
I´m trying to solve this integral, but I don't know how use the power representation in this integrals.
Let $\gamma(t)=e^{it}$ for $t\epsilon[0,2\pi]$
$$\int_{\gamma }\frac{(e^{z}-e^{-z})}{z^...
0
votes
0answers
49 views
Is there an upper Bound for $s_n := \sum_{k=0}^n a_k > 0$ knowing $\limsup_{k\rightarrow \infty} \sqrt[k]{|a_k|} = 1$
Given $s_n := \sum_{k=0}^n a_k$ and $s_n > 0$ for every $n\in\mathbb{N}$ and $\limsup_{k\rightarrow \infty} \sqrt[k]{|a_k|} = 1$. Determine, whether there is an upper bound for $s_n$.
I am quite ...
0
votes
1answer
29 views
Is the series expansion of $\frac1{(1+x)^n}$ same as $\frac1{(1-x)^n}$ with $(-1)^r$
The series expansion for $$\frac1{(1-x)^n} =
\sum_{r=0}^{r=\infty}C_r^{|n|+r-1}x^r$$
Is the expansion of $$\frac1{(1+x)^n} = \sum_{r=0}^{r=\infty}(-1)^rC_r^{|n|+r-1}x^r$$ ?? (Where C is combination)
3
votes
3answers
102 views
what is the general way of proving that a series converges uniformly on a bounded interval?
I am trying to solve the following question:
Prove that the series $$\sum_{n = 1}^{ \infty} \frac{x^{2n - 1}}{(2n - 1)!} $$ and the series $$\sum_{n = 1}^{ \infty} \frac{x^{2n}}{(2n)!}$$ both ...
0
votes
3answers
38 views
Struggling to compute a power series for a complex value function
I am struggling to compute the power series expansion of $$f(z) = \frac{1}{2z+5}$$ about $z=0$, where $f$ is a complex function. I tried comparing it to the geometric series as follows,$$ f(z) = \frac{...
4
votes
1answer
161 views
Formulas for $\pi$ of the form $2\sum_{k=0}^\infty\binom{2k}{k}\frac{a^{2k+1}+b^{2k+1}+c^{2k+1}}{4^k(2k+1)}$
Third edit:
For those interested in the Sagemath-code to produce your own formula, given three natural numbers $x<y<z$, it can be found here. I am sharing those formulas in public domain, for ...
1
vote
1answer
65 views
Simplifying $\sum_{n=1}^{\infty}\frac{n\alpha^n}{(n-m)!(n+z)}(x-c)^{n-m}$
We would like to simplify the following summation
$$\sum_{n=1}^{\infty}\frac{n\alpha^n}{(n-m)!(n+z)}(x-c)^{n-m}.$$
What we know is that $x, z, \alpha, c > 0$ and $z$ is integer.
Here's what I did ...
2
votes
1answer
49 views
Generating Function Calculation Gone Wrong
Let $T$ be a function of two complex variables, analytic at the origin. Express $T$ as a power series:
$$T(x,y)=\sum_{i,j}t_{ij}x^iy^j$$
I would like to find a generating function for the sequence $(...
0
votes
0answers
35 views
Trying to understand / apply the Hilbert–Serre theorem in a specific situation?
In A series related to prime numbers the function $H(t)$ is defined with a hint, that it is a Hilbert–Poincaré series of $V := \bigoplus_{n\in \mathbb{N}} V_n $, where $V_n := \left\langle \log(1), \...
-1
votes
0answers
20 views
Power series as analitycs functions
I am searching how to prove that a power series is an analityc function because it is something that I have studied but without a prove.
0
votes
1answer
35 views
Suppose $\sum_{i = 1} a_i \sum_{i = 1} b_i < \infty$. If $\sum_{i = 1} a_i = \infty$, what does it say about $(b_i)$?
Let $(a_i), (b_i)$ be two non-negative sequence.
Suppose $\sum_{i = 1} a_i \sum_{i = 1} b_i < \infty$. If $\sum_{i = 1} a_i = \infty$, what does it say about $(b_i)$?
Does it necessarily mean that ...
0
votes
0answers
38 views
Power series interval of convergence $ x^k$ multiplication/division
If we have a power series and we multiply it by $x^k$ where $k$ is fixed natural number, does the resulting series have the exact same interval of convergence?
Also if we have power series starting ...
2
votes
3answers
62 views
Maclaurin series of $(1+x^3)/(1+x^2)$
I can't seem to figure out the Maclaurin series of $(1+x^3)/(1+x^2)$
I started with $ 1/(1-x)= \sum_{n=1}^{\infty} x^n $
$ 1/(1-(-x^2)) = \sum_{n=1}^{\infty}(-1)^n x^{2n} $
$ (1+x^3) \sum_{n=1}^{\...
1
vote
1answer
154 views
An upper bound for the function $x!$ using some well-know constant as $e$ or $\pi$
Problem :
Define :
$$f(x)=\left(e^{\frac{3x\left(\pi^{x}-e^{x}\right)}{\pi^{x}+e^{x}}}-e^{\sqrt{2x}-3}\right)^{\frac{3}{\pi}}$$
Let $x>\frac{1}{10}$ then prove or disprove that :
$$j(x)=f\left(x\...
1
vote
0answers
53 views
$2y'' + (x + 1)y' + 3y = 0$ with power series
I am trying to solve this ODE with power series.
$$2y'' + (x + 1)y' + 3y = 0$$
$$x_0=2$$
since $x_0$ is an ordinary point, the answer we guess is:
$$y=\sum_{n=0}^{\infty}a_n(x-2)^{n}$$
we get:
$$y'=\...
0
votes
0answers
36 views
Value of series involving central factorial numbers and zeta function
I encountered an interesing series and was wondering if its value can be computed.
First we consider the central factorial numbers. For $n\in \mathbb N$ we define the polynomial
$$
P_n(x) = x(x + \...
0
votes
1answer
55 views
Laurent Series of $f(z) = \frac{(z+1)^2}{z(z^3+1)}$ about $z = 0$?
This is supposed to be a non-calculator question, so I managed to get this far;
$ z^3 + 1 = (z + 1)(z^2 - 1 +1)$ , by polynomial division.
Therefore, $ f(z) = \frac{z+1}{z(z^2-z+1)} = (1 + \frac{1}{...
0
votes
2answers
82 views
About the inequality $\left(\ln\left(\frac{4+x}{2x+2}\right)\right)^{2}\leq x!$
Problem :
Let $-1<x<0$ then we have :
$$\left(\ln\left(\frac{4+x}{2x+2}\right)\right)^{2}\leq x!\tag{I}$$
My attempt:
I have tried :
$$x!=\frac{1}{x+1}-\gamma+O(x^2)$$
Wich is not enought .
...
3
votes
1answer
94 views
Power series of $\frac{x+1}{x^2+5}$
I have this problem: write down the power series of $$\frac{x+1}{x^2+5}$$ and specify for which interval the equality (function = series) holds true.
I noticed that I can use differentiation method ...
0
votes
1answer
65 views
Prime consequences [closed]
During my research about zerouilia consequence here I guessed about the nature of the following consequences of prime numbers
$U_p=\frac{1}{2}^{{\frac{1}{3}}^{{\frac{1}{5}}...^{\frac{1}{p}}}}$ and ...
6
votes
2answers
72 views
Closed form solution method for infinite sum (but not "by inspection")
I have a series whose terms are defined by the recurrence relation
$$T_n = \alpha \ \left(1+\frac{\mu}{n}\right)\ T_{n-1}$$ with $T_0=1$
So $$T_n=\alpha^n\prod_{k=1}^n \left(1+\frac{\mu}{k}\right)\tag{...
0
votes
1answer
83 views
Can we write a series based on its first 5 terms?
We are interested in formulating a function of $x$ as a series.
We know that, for example, the first five elements are
\begin{align}
i=1 &\rightarrow \frac{1}{(-1)^1(x-c)^1} \\
i=2 &\...
1
vote
0answers
53 views
Continuity at the boundary of a convergent power series
Say $f(z)=\sum a_{n}z^{n}$ is a power series with convergence radius $0<R<\infty$. Suppose we know that the series convergence at $z_{0}$ where $z_{0}$ is a point at the boundary of the ...
4
votes
0answers
90 views
Proving $\int_{0}^{1}\frac{\ln(x)}{1+x}\,\mathrm{d}x = \sum\limits_{n=1}^{+\infty}\frac{(-1)^n}{n^2}$ using uniform convergence theorem
Prove that
$$\int_{0}^{1}\frac{\ln(x)}{1+x}\,\mathrm{d}x = \sum_{n=1}^{+\infty}\frac{(-1)^n}{n^2}$$
using uniform convergence theorem.
I already know that you can obtain quickly this equality using ...
0
votes
0answers
34 views
Understanding a non-linear polynomial recursion relationship.
I am trying to find a solution to a non-linear PDE. The solution $w(x,y)$ solves
$$
\partial_y^2w+\partial_x(\exp(2w)\partial_x w)=0
$$
and I am interested in a solution of the form
$$
w=\frac{1}{2}\...
1
vote
1answer
82 views
Questions concerning analytic and singularities of power series
In the book 'The Concrete Tetrahedron' by Manuel Kauers and Peter Paule on p30 reads or implies the function
$$f: C\backslash \{ ... ,-2\pi,-\pi,0,\pi,2\pi, ...\}\rightarrow C,\, f(z)=\frac z{\sin(z)}$...
0
votes
0answers
35 views
Developping Gamma Function in power series
Let take the following function, which is similar to the Gamma Function:
$$
f(x) = \int_{0}^{\infty }t^{x^{2}-1}e^{-2t}dt
$$
How could we develop this function in power series?
I try to develop it ...
0
votes
0answers
55 views
Where is an expository article on proving Newton's binomial theorem?
Some years ago I found an expository article devoted to proving Newton's binomial theorem, which says that if $n$ is any real (not necessarily positive and not necessarily integral) number, and $|x|&...
1
vote
1answer
121 views
Show that $\lim _{x\rightarrow 1}\frac{\sum _{n=0}^{\infty }a_nx^{n\ }}{\frac{1}{1-x}\log \left(\frac{1}{1-x}\right)} = 1$
The problem is stated as:
Given that $(a_n)_{n=1}^{\infty}$ is a sequence for which $a_n/\log(n) \rightarrow 1$ as $n\rightarrow \infty$, show that $$\lim _{x\rightarrow 1}\frac{\sum _{n=0}^{\infty }...
1
vote
1answer
57 views
Prove that the radius of derived series $\sigma'$ is the same of $\sigma$.
Let be $\sigma$ the power series given by the equation
$$
\sigma:=\sum_{k=0}^\infty a_k x^k
$$
so that we call radius convergence of $\sigma$ the quantity $\rho$ given by the equation
$$
\rho:=\sup\{x\...
3
votes
1answer
88 views
$(1+x)^a$ McLaurin series
My book has a small section about the Maclaurin series of $f(x) = (1+x)^a$.
We look at this function only when $x \gt -1$. So the domain is just $(-1, \infty)$
The number $a$ is a real constant (could ...
0
votes
0answers
24 views
efficient (non series expansion ?? ) computation of log(P / 1 - P)
As per the question, where P is a probability $[0,1]$.
The reason for seeking optimisation is that my system will have to
perform $O(10^9)$ such computations per second.
I suspect that a rearrangement ...
1
vote
1answer
52 views
Power series of $x/(1-ae^{-x})$.
I am looking for a power series expansion for the function $x(1-ae^{-x})^{-1}$ (perhaps for $0<a<1$).
Using the Bernoulli numbers, I can write
\begin{align*}
\frac{x}{1-ae^{-x}} &= \frac{x}{...
2
votes
0answers
69 views
Large $x$ power series for $e^{-\alpha x}$
Let $\alpha>0$, then do there exist continuous functions $c_n\colon\mathbb R^+\to\mathbb R$ and $\gamma\colon\mathbb R^+\to\mathbb R$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$ and the ...
2
votes
1answer
95 views
Do you have equipment with the computational accuracy to test my infinite series representation for Gamma(m), or (m-1)! ….?
I have derived the following alternating infinite series as an approximation to values of the Gamma Function which yields reasonably accurate results between Gamma(1) and Gamma(7)-ish. Beyond that, ...
0
votes
0answers
13 views
Is there a natural, operations-preserving bijection between Levi-Civita field and (a subset of) divergent integrals?
For instance, would the following definition of multiplication on divergent integrals correspond to the multiplication in Levi-Civita field?
$\int_0^\infty f(x)dx \cdot \int_0^\infty g(x)dx =\int_0^{\...
1
vote
2answers
104 views
Proving $\prod_{n=0}^{\infty}\left(1+\frac{x}{a^n}\right)=\sum_{n=0}^{\infty}\frac{(ax)^n}{\prod_{k=1}^{n}(a^k-1)}$
By trying to prove that Riemann's Zeta function is analytically expendable to the whole plane with one pole, I went aside and noticed this identity about formal power series (which are obviously ...
1
vote
1answer
47 views
Can $f(x)=\int_{0}^{\infty}t^{x^2+1}e^{-2t}dt$ be written as a power series in a neighbourhood of zero?
Let $f(x)=\int_{0}^{\infty}t^{x^2+1}e^{-2t}dt$. Can $f(x)$ be written as a power series in a neighbourhood of zero? In this case, what would its convergence radius be?
Trying to solve this problem I ...
0
votes
0answers
35 views
Formal Taylor Series Expansion
In this paper, the authors consider how to convert a system of $N$ ODEs to a PDE where $N \rightarrow \infty$ in some appropriate sense. In what follows $u_{n+1}$, the solution of the $(n+1)$th ODE, ...
26
votes
0answers
359 views
An iterative logarithmic transformation of a power series
Consider the following iterative process. We start with the function having all $1$'s in its Taylor series expansion:
$$f_0(x)=\frac1{1-x}=1+x+x^2+x^3+x^4+O\left(x^5\right).\tag1$$
Then, at each step ...
2
votes
1answer
38 views
Continuation of an asymptotic series originally defined for $z>0$ to $z<0$
Consider the well-known asymptotic expansion of the $\Gamma%$-function:
$$\Gamma(x)\sim\left(\frac{x}{e}\right)^x \sqrt{\frac{2 \pi }{x}}\left[1+\frac{1}{12 x}+\frac{1}{288 x^2}-\frac{139}{51840 x^3}-\...