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.

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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{...
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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 $$f_n(x)=\frac{\partial^n}{\partial y^n}e^{xy}|_{y=0}$$
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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
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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
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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} = \...
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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, ...
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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 $f(x)=2^{-x^2}\Gamma(x^2)$ ...
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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 ...
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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(...
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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^...
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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 ...
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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
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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 ...
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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{...
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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 ...
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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 ...
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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 $(...
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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), \...
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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.
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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 ...
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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 ...
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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}^{\...
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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\...
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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'=\...
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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 + \...
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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}{...
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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
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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 ...
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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
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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{...
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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 &\...
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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 ...
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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 ...
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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}\...
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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)}$...
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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 ...
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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|&...
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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
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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
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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 ...
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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
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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
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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
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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, ...
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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^{\...
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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
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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
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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
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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
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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}-\...

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