Skip to main content

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.

Filter by
Sorted by
Tagged with
3 votes
2 answers
75 views

I need help to prove that$\int^1_0 x^{m-1}\ln^2(1-x)dx=\frac{2}{m}\sum_{k=1}^m\frac{H_{k}}{k}$

Show that $$\int^1_0 x^{m-1}\ln^2(1-x)dx=\frac{2}{m}\sum_{k=1}^m\frac{H_{k}}{k}$$ We know: $$\ln(1-x)=-\sum^{\infty}_{n=1}\frac{x^n}{n}$$ and using cauchy product Therfore: $$\begin{align*} \ln^2(1-x)&...
Mostafa's user avatar
  • 2,322
-1 votes
0 answers
39 views

I need to convert $\frac{x-\arctan x}{x^2}$ into a power series, and then integrate it. The series should start at $n=1$. [closed]

I need to convert $\frac{x-\arctan x}{x^2}$ into a power series, and then integrate it. The series should start at $n=1$.
FChikers's user avatar
-2 votes
2 answers
97 views

An inequality for arctan in terms of hyperbolic tangent and radicals

Problem : Let $x>0$ then define : $$h\left(x\right)=\arctan\left(x\right)-\tanh\left(x\right)-\frac{1}{7}\left(1-\tanh\left(\frac{1}{x}\right)\right)^{\frac{3}{2}}-\frac{1}{3}\left(1-\tanh\left(\...
Ranger-of-trente-deux-glands's user avatar
1 vote
1 answer
81 views

Find the Laurent expansion of $f(z) = \sin{\frac{1}{z(z-1)}}$ in $0<|z-1|<1$

Here is my idea: $\sin{\frac{1}{z(z-1)}} = \sin{\left( \frac{-1}{z} + \frac{1}{z-1}\right)} = \sin{\left(\frac{1}{z-1} - \frac{1}{z}\right)} = \sin{\frac{1}{z-1}}\cos{\frac{1}{z}} - \cos{\frac{1}{z-1}}...
Irbin B.'s user avatar
  • 172
1 vote
0 answers
61 views

How to find the roots of sin(x) using series theory

If we define $$\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n \ x^{2n+1}}{(2n+1)!}$$ How to find the roots of $\sin(x)$, i.e. $$\pi =4 \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$$ satisfies $\sin (\pi)=0$
Phy-zr's user avatar
  • 21
0 votes
1 answer
54 views

Counter example to the identity theorem for two generating functions

I want to give an example of two generating functions $\psi_{X_+}$ and $\psi_{X_-}$ for random variables $X_+$ and $X_-$ with values in $\mathbb{N}_0$ which coincide on infinitely many points $x_i\in(...
Christoph Mark's user avatar
5 votes
0 answers
101 views

Zeta Lerch function. Proof of functional equation.

so I'm trying to prove the functional equation of Lerch Zeta, through the Hankel contour and Residue theorem, did the following. In the article "Note sur la function" by Mr. Mathias Lerch, a ...
Nightmare Integral's user avatar
2 votes
3 answers
147 views

Infinite Series : $1+\frac{1}{3^3}-\frac{1}{5^3}-\frac{1}{7^3}+\ldots$

I need Help to evaluate infinite series : $$S=1+\frac{1}{3^3}-\frac{1}{5^3}-\frac{1}{7^3}+\cdot\cdot\cdot$$ My try: Let $$c_n:= \left({\frac{1-i}{2}}\right)(i)^n+\left({\frac{1+i}{2}}\right)(-i)^n$$ ...
Mostafa's user avatar
  • 2,322
1 vote
0 answers
33 views

Mellin transform of confluent Lauricella hypergeometric function

The $F_D^{(n)} $ Lauricella's hypergeometric function can be defined as follow $$F_D\left(a,b_1,\cdots,b_n;c;x_1,\cdots,x_n\right) = \sum_{m_1=0,\cdots,m_n=0}^{\infty}\frac{\left(a\right)_{m_1+\cdots+...
K.K.McDonald's user avatar
  • 3,263
2 votes
1 answer
69 views

I need Help to evaluate :$\sum_{n=0}^{\infty} \left({\frac{(2n+1)!!}{(2n+2)!!}}\right)^2\frac{1}{(2n+4)^2}$

I need Help to evaluate :$$S=\sum_{n=0}^{\infty} \left({\frac{(2n+1)!!}{(2n+2)!!}}\right)^2\frac{1}{(2n+4)^2}$$ we have : $$\int^{\frac{\pi}{2}}_0\cos^{2n+2}(x)dx=\int^{\frac{\pi}{2}}_0\sin^{2n+2}(x)...
Mostafa's user avatar
  • 2,322
0 votes
0 answers
26 views

A problem of determining whether a power series belongs to $\mathbb{C}(u)$

I am reading a paper "Drinfeld coproduct, quantum fusion tensor category and applications" and I have a probelm. Here is the arxiv:Drinfeld coproduct, quantum fusion tensor category and ...
fusheng's user avatar
  • 1,159
2 votes
1 answer
72 views

I need Help to evaluate series :$\sum_{n=0}^{\infty} \frac{(2n+1)!!}{(2n+2)!!}\frac{1}{n+1}$

I need Help to evaluate series :$$\sum_{n=0}^{\infty} \frac{(2n+1)!!}{(2n+2)!!}\frac{1}{n+1}$$ Let :$u_n=\frac{(2n+1)!!}{(2n+2)!!}\frac{1}{n+1}$ We have $$\lim_{n\to\infty} n\left({\frac{u_n}{u_{n+1}}-...
Mostafa's user avatar
  • 2,322
0 votes
1 answer
43 views

Proof of radius of convergence

For all $|x|\lt1, \displaystyle\sum_{n=1}^{\infty}a_nx^n$ converges. I need to prove that $\displaystyle \sum_{n=1}^{\infty} a_nx^{2n+1}$ converges for all $|x|\lt1$. Is my proof correct? For all $|x| ...
talopl's user avatar
  • 1,009
2 votes
3 answers
103 views

Find the domain of convergence of $\sum\limits_{n=1}^{\infty} (e - (1+\dfrac{1}{n})^n)^{2x}$

I would like to find the domain of convergence of the series $\sum\limits_{n=1}^{\infty} \left(e - \left(1+\dfrac{1}{n}\right)^n\right)^{2x}$. In fact, I knew that $\lim \left(e - \left(1+\dfrac{1}{n}...
Mariod's user avatar
  • 71
0 votes
0 answers
23 views

number of zeros of a power series defined by an absolutely convergent sequence

Let $a = (a_n)_{n \in \mathbb{N}}$ be an absolutely convergent sequence such that each $a_n \in \mathbb{R}$, and define $f_a(x) = \sum_{n \in \mathbb{N}} a_n x^n$ for $x\in\mathbb{R}$. To avoid a ...
hs12's user avatar
  • 1
3 votes
1 answer
154 views

Why can we convert a power series of operators to a function, invert the function, and then work with its series expansion?

I apologize as I am still somewhat unfamiliar with infinite series and knowing when and how we are allowed to use them, and especially with working with operators in this way. I imagine that I will be ...
Vipinx's user avatar
  • 35
4 votes
3 answers
70 views

$(1-x)^{n+a} \sum_{j=0}^\infty \binom{n+j-1}{j}\binom{n+j}{a} x^j = \sum_{j=0}^a \binom{n}{a-j}\binom{a-1}{j} x^j$

Let $n$ and $a$ be natural numbers. How to prove the following for $x \in [0, 1)$? $$ (1-x)^{n+a} \sum_{j=0}^\infty \binom{n+j-1}{j}\binom{n+j}{a} x^j = \sum_{j=0}^a \binom{n}{a-j}\binom{a-1}{j} x^j $$...
ploosu2's user avatar
  • 9,738
0 votes
1 answer
83 views

$ (1 + z^2)w'' - 2w = 0 $ in complex space

Find the linearly independent solutions of the equation $(1 + z^2)w''-2w = 0$ in the vicinity of the point $0$. Attempt: To find the linearly independent solutions to the differential equation: $(1 + ...
user avatar
0 votes
0 answers
33 views

Maximizing the radius of convergence around a point for an analytic function

Let $f:\Omega\longrightarrow \mathbb{C}$ be analytic, and let $z_0\in\Omega$ s.t. $$f (z)=\sum_{n\geq 0} a_n(z-z_0)^n,\quad\forall \left|z-z_0\right|<r,$$ for some $r>0$, and some complex-valued ...
virtualcode's user avatar
0 votes
1 answer
65 views

I need help to evaluate series :$\sum^{\infty}_{n=0} \frac{(n+1)^{n-1}}{n!}(xe^{-x})^n$

I need help to evaluate series :$$S(x)=\sum^{\infty}_{n=0} \frac{(n+1)^{n-1}}{n!}(xe^{-x})^n$$ My attempt was to find an integral equal to $\frac{(n+1)^{n-1}}{n!}$ But I couldn't find it This is what ...
Mostafa's user avatar
  • 2,322
1 vote
1 answer
82 views

Show that $\frac{1+z-\sqrt{z^2-6z+1}}{4}$ fits the Lagrangean framework

Let $S(z)$ be the OGF of bracketings. Show that the Lagrangean framework holds for $S(z)$. Remark: You can find the definition of Lagrangean framework below. From the Flajolet & Sedgewick book (p....
3nondatur's user avatar
  • 4,212
0 votes
0 answers
31 views

Show that $\partial ^\beta f(x)=\sum _{\alpha\in\mathbb{N}^n}c_{\alpha +\beta }\frac{(\alpha +\beta )!}{\alpha!}(x-x_0)^\alpha$

Firstly consider the multi-index notation. Let $\{c_\alpha\}_{\alpha\in\mathbb{N}^n}\subseteq\mathbb{R}$ and $x_0:=(x_{01},\cdots,x_{0n})\in\mathbb{R}$. Define $\rho :=\sup\big\{r\in [0,\infty ):\sum_{...
rfloc's user avatar
  • 1,171
3 votes
2 answers
217 views

Power series representation of $1/(2+x)$

Why the power series representation of $\frac{1}{2+x}$ is $$\sum_{n=0}^{\infty}\frac{(-1)^n}{2^{n+1}} x^n$$ and not $$\sum_{n=0}^{\infty}(-1-x)^n$$ based on $\frac{1}{1-x}$ = $\sum_{n=0}^{\infty}x^n$ ...
Darren's user avatar
  • 31
1 vote
0 answers
83 views

Problem 6.2.10 From Complex Analysis by Jihuai Shi

Definition: let $f$ be holomorphic on a region $G$ (here region means a non-empty connected open set). For $\xi\in \partial G$, if there exists a ball $B(\xi,\delta)$ and a holomorphic function $g \in ...
Robert's user avatar
  • 11
1 vote
1 answer
46 views

Taylor-Laurent series expansions

I'm having some issues finding how to series expand some complex functions that my professor gave past years in exams. For example, in this exercise, it is asked to find the first two terms of the ...
deomanu01's user avatar
  • 113
1 vote
0 answers
82 views

Neumann series expansion for the resolvent at the spectral radius

Let $T$ be a bounded linear operator on a Banach space. We have following formula for the resolvent: $$\frac{1}{\lambda I-T}=\frac{1}{\lambda }\frac{1}{I-\frac{T}{\lambda }}=\frac{1}{\lambda }\left(I+\...
Damalone's user avatar
  • 329
0 votes
0 answers
28 views

Deriving a three term recurrence relation from a DE using the Frobenius Method

I am trying to derive a three-term recurrence relation from the DE given by the equation below: $$ \frac{d^2R(r)}{dr^2} + \frac{2}{r}\frac{dR(r)}{dr} + \left(2E - V(r) - \frac{l(l+1)}{r^2}\right)R(r) =...
Benoni defence's user avatar
3 votes
1 answer
62 views

Generating functions of Wallis integrals

simple question, how would one go about proving that $$ \sum_{n\ge 0}^{}W_{n}x^{n}=\int_{0}^{\frac{\pi}{2}}\frac{dt}{1-x\cos t} $$ from this, it results with substituing $u= \tan(t/2)$ that $$ \sum_{n\...
realreal's user avatar
  • 125
2 votes
1 answer
62 views

Solving $y''(x)-2xy'(x)+y(x) = 0$ using power series

I'm solving $y''(x) -2xy'(x) +y(x) = 0$ using the power series ansatz $y(x) = \sum_{n=0}^{\infty}a_n x^n$ . Plugging in I get: \begin{equation} \bigg(\sum_{n=0}^{\infty}n(n-1)a_nx^{n-2}\bigg)-2x\bigg(\...
haifisch123's user avatar
2 votes
0 answers
32 views

Definition and Use of the Schett Polynomial in the Jacobi Taylor Series

I am having a tough time understanding the definition and use of the Schett polynomial introduced in the paper here. I have two questions related to this polynomial. My first question concerns its ...
Kyler Rusin's user avatar
1 vote
1 answer
65 views

How to calculate the generalized Puiseux series of ${\csc(x)}^{\tan(x)}$ at $x=0$?

First consider this problem: find the limit of ${\csc(x)}^{\tan(x)}$ as $x\rightarrow0$, which is easy: $$ \begin{aligned} \lim_{x\to0}{\csc(x)}^{\tan(x)}&=\lim_{x\to0}\exp(\tan(x)\ln(\csc(x)))\\ &...
El Mismo Sol's user avatar
3 votes
1 answer
84 views

Taylor expansion of $x^2/(k+x^2)$

I'm trying to write down a formula for the Taylor expansion of $x^2/(k+x^2)$ around $x=z$. The Taylor expansion for $x/(k+x)$ around $x=z$ is \begin{align*} \frac{x}{k+x} = \frac{z}{k+z} + k \sum_{j=1}...
tomjonson's user avatar
0 votes
0 answers
32 views

What is the power series for $K_{\lambda}(w) = \int_{0}^{\infty}z^{\lambda-1}\exp\left\{-\frac{w}{2}\left(z + \frac{1}{z}\right)\right\}\mathrm{d}z$?

As the title says, I would like to know what is the power series for (or at least a good approximation) \begin{align*} K_{\lambda}(w) = \frac{1}{2}\int_{0}^{\infty}z^{\lambda-1}\exp\left\{-\frac{w}{2}\...
learner123's user avatar
2 votes
0 answers
26 views

Reference request for Bessel function of the second kind with matrix argument

As the title says, I would like to know if anyone could provide a reference which provides the definition and properties of the Bessel function of the second kind with matrix argument. If possible, I ...
learner123's user avatar
2 votes
1 answer
39 views

Uniform convergence of $\sum_{n=1}^{\infty}x^n\sqrt{1-x}=\sum_{n=1}^{\infty} f_n(x)$ on $(0,1)$

I want to check whether this series of functions converges uniformly or not. $\textit{Attempt:}$ I have used Weierstrass M-Test to find the maximal value for $f_n(x)$ and I got $M_n=(\frac{2n}{2n+1})^...
Bowei Tang's user avatar
  • 1,513
0 votes
0 answers
55 views

Show that the function $f(x)=\begin{cases} e^{-1/x^2} & x\neq 0 \\ 0 & x=0 \end{cases}$ has no Maclaurin's series [duplicate]

Show that the function $$f(x)=\begin{cases} e^{-1/x^2} & x\neq 0, \\ 0 & x=0 \end{cases}$$ has no Maclaurin's series expansion but the function has derivative of all order everywhere. We have $...
user1942348's user avatar
  • 3,903
0 votes
2 answers
58 views

Using Ratio Test for a Series with odd and even indexed coefficients

The problem I have is the following: Suppose the radius of convergence of the series $\sum_{n=0}^\infty a_n z^n$ is equal to $2$. Find the radius of convergence of the series $$\sum_{n=0}^\infty 2^{\...
Hyperbolic Cake's user avatar
4 votes
3 answers
145 views

How to Find closed form $\sum_{n=1}^{\infty}\frac{x^{2n}}{n}a_n$

How to Find closed form :$$\sum_{n=1}^{\infty}\frac{x^{2n}}{n}a_n$$ where $a_n=\sum_{k=1}^{2n-1}\frac{(-1)^{k-1}}{k}$ $$S(x)=\sum_{n=1}^{\infty}\frac{x^{2n}}{n}a_n=\sum_{n=1}^{\infty}\int^1_0y^{n-1}x^{...
Mostafa's user avatar
  • 2,322
0 votes
0 answers
21 views

Uniqueness of Two Series in an Intersection

I've been working on some problems involving series and have found myself applying the identity principle to show that two representations of a series will lead them being unique under some conditions....
Hyperbolic Cake's user avatar
1 vote
1 answer
35 views

Frobenius method for solving differential equation

Need help solving this differential equation$$xy''+(x+l)y'+ly=0$$ where $l\in \mathbb{R}$\ I tried using the Frobenius method. $x_0=0$ is a regular singular point and Assume that we have the solution ...
anon02001's user avatar
1 vote
0 answers
32 views

Taylor Approximation of Third Order

Problem: Given the function $ f(x, y) = e^{x^2} \log(1 + x + y) $ near the point $ (0, 0) $, find the third-order Taylor approximation of the function at $ (0, 0) $ using known series and verify the ...
j.primus's user avatar
4 votes
3 answers
159 views

Where is the error in evaluating this series? $\sum_{n=1}^{\infty}\frac{1}{n(4n^2-1)^2}$

Where is the error in evaluating this series? $$\sum_{n=1}^{\infty}\frac{1}{n(4n^2-1)^2}$$ We have $$ \frac{1}{n(4n^2-1)^2} = \frac{1}{n} - \frac{1}{2n-1} - \frac{1}{2n+1} + \frac{1}{2(2n-1)^2} - \...
Mostafa's user avatar
  • 2,322
2 votes
2 answers
116 views

How can to find to infinite series ? $\frac{1}{x} -\frac{1}{1!}\frac{1}{x+1}+\frac{1}{2!}\frac{1}{x+2}-\frac{1}{3!}\frac{1}{x+3}+...+\infty$

How can I continue to find a solution to infinite series ? $\frac{1}{x} -\frac{1}{1!}\cdot\frac{1}{x+1}+\frac{1}{2!}\cdot\frac{1}{x+2}-\frac{1}{3!}\cdot\frac{1}{x+3}+\cdot\cdot\cdot+\infty$ we have $$...
Mostafa's user avatar
  • 2,322
2 votes
0 answers
71 views

How can this series be calculated? $\frac{1}{1+x}+\frac{2}{1+x^2} + \frac{4}{1+x^4}+\frac{8}{1+x^8} +\cdot\cdot\cdot $ [duplicate]

How can this series be calculated? $$\frac{1}{1+x}+\frac{2}{1+x^2} + \frac{4}{1+x^4}+\frac{8}{1+x^8} +\cdot\cdot\cdot $$ we have : \begin{align} S(x) & =\frac{1}{1+x}+\frac{2}{1+x^2} + \frac{4}{1+...
Mostafa's user avatar
  • 2,322
1 vote
1 answer
35 views

What function can model a decay, whose linear slope smoothly changes around a certain x value.

am a biology student trying to find a fitting function to model the $1/f$ decay or 'aperiodic component' of a neural power spectrum. In an (over)simplified way it can often be describes with the ...
Stav32's user avatar
  • 11
3 votes
1 answer
108 views

Is there a name for this formula for central trinomial powers?

I have determined (by applying the techniques found in this blog post) that if $a_n$ is determined by the $n$th coefficient of $(ax^2+bx+c)^n$, then the generating function of $a_n$ is $$\frac1{\sqrt{(...
Akiva Weinberger's user avatar
1 vote
0 answers
72 views

Power series expansions and limits of knot invariants

I move the question here Background(ignore this paragraph if you know finite type invariants well): Recall that a finite type invariant of degree $n$ is an invariant $V$ such that $V^{(n+1)}=0$ where $...
Eric Ley's user avatar
  • 738
1 vote
0 answers
63 views

How to evaluate This Series :$\dfrac{x}{1+x}+\dfrac{2x^2}{1+x^2}+\dfrac{4x^4}{1+x^4}+\cdots$ [duplicate]

How to evaluate This Series :$$S=\dfrac{x}{1+x}+\dfrac{2x^2}{1+x^2}+\dfrac{4x^4}{1+x^4}+\cdots$$ therfore: $$S=\dfrac{x}{1+x}+\sum_{k=1}^{\infty}\dfrac{2kx^{2k}}{1+x^{2k}}$$ This series Converges if $...
Mostafa's user avatar
  • 2,322
2 votes
1 answer
61 views

Counting irreducible Permutations

Let $x:=(x_1,...,x_n)$ be a permutation of $\{1,...,n\}$. We say, $x$ is irreducible iff $\{x_1,...,x_m\}\neq\{1,...,m\}$ for $1\leq m \leq n-1$. Let $g(n)$ be the number of irreduible permutations of ...
NTc5's user avatar
  • 609
1 vote
1 answer
85 views

Inverse Series of $x\sin x$

For some background before I get into my question: I am a Calculus 2 student who knows only some of the bare essentials to Complex Analysis, so bear with me. I was recently studying the transformation ...
Oiler's user avatar
  • 33

1
2 3 4 5
172