Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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7 views

rewriting product of power series

According to $$\Lambda(\tau;q)=B_0(\tau)+\sum_{i=1}^\infty B_i (\tau) q^i$$ we define $$[\Lambda(\tau;q)-B_0(\tau)]^m=\bigg[ \sum_{i=1}^\infty B_i (\tau) q^i ...
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2answers
61 views

Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation

Find sum of the series $$\sum_{n=1}^{\infty}\frac{x^{3n}}{\left(3n\right)!}$$ using differentiation. So far I found that $$S(x)+1=S'''(x)$$ but it does not help. Then I tried different interesting ...
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0answers
29 views

How do I get these equations ($n$th-order deformation equation and its initial/boundary conditions)?

Earlier, I asked a question What does 'equating the like-power of $q$ ' mean? and I already got the answer about the meaning of a particular phrase in a certain context. However, my real question ...
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2answers
112 views

Power series expansion of $x\ln(\sqrt{4+x^2}-x)$

Find $a_n $ where $x \ln(\sqrt{4+x^2}-x) =\sum_{n=0}^{\infty} a_nx^n$. I know that I must find power series expansion of $\ln(\sqrt{4+x^2})$ but it doesn't help. Can anyone give me a hint? many ...
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1answer
21 views

Reciprocal of power series with same radius

Let $f$ be a power series $f(x)=\sum a_n x^n$ with radius $R=\limsup \frac{1}{(\sqrt{|a_n|})^\frac{1}{n}}$ defined in $]-R,R[$. Let us suppose that $|f(x)|>c$ for a given $c$. Claim: Its ...
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1answer
29 views

Coefficients in power series of $\frac{3x-1}{x^3+3x-1}-\frac1{6(1-2x)}$ is bounded

Suppose that $\frac{3x-1}{x^3+3x-1}-\frac1{6(1-2x)} = \sum_{n\geq 0}c_nx^n$. Show that $|c_n|$ is bounded by some constant. I have no clue to prove the claim. Can anyone give me some hints? Thank ...
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1answer
22 views

Problem with the inverse expansion

Let $q=e^{2\pi i z}$ and $t=q-12q^2+66q^3-220q^4+495q^5-...$ Then why is the inverse expansion equal to $q=t+12t^2+222t^3+...$? I also do not understand the notation here: $t$ means $t(z)$ or $t(q)$? ...
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1answer
24 views

What does 'equating the like-power of $q$' mean?

I am reading a book "Homotopy Analysis Method in Nonlinear Differential Equations" by Shijun Liao chapter 13 Applications in Finance: American Put Options. It is stated there that Substituting ...
2
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1answer
79 views

How to find the bound of this sum?

Let $t>0,a(t)=\arg(\Gamma(1/4+it))$,$\kappa(n)=\frac{1}{2}x\pi n^2$,we need to calculate the bound,$A(x)$, of the following finite sum: $$ S(x)=\sum_{1\le n\le ...
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29 views

Laurent series of $f(z)=\frac{1}{z(z-1)(z-2)}$

I am trying to find the Laurent series of the function $$f(z)=\frac{1}{z(z-1)(z-2)}$$in the rings: 1) $0<|z-1|<1$, 2) $1<|z-1|$, 3) $1<|z-2|<2 $ First I expressed $f$ as ...
2
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0answers
26 views

function with branch cuts : the radius of convergence of its Taylor series

Let $f(z)$ being the analytic continuation of some holomorphic function, having many branch points and isolated singularities at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$ is the radius of convergence ...
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1answer
26 views

What is the Laurent expansion of $f(z)=\frac1{z-3}$?

What is the Laurent expansion of $f(z)=\dfrac1{z-3}$? In the region, $|z-3|>0$ ? I just computed the Laurent expansion in the region $|z|>3$ by dividing the denominator by $\dfrac1z$ and ...
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1answer
702 views

Equation about generating functions and subfactorial

Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following explanation) of variable $w$, try to solve out $G_n(w)$ for all $n\ge0$ from the ...
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2answers
82 views

Proving complex trigonometric identity using power series

Prove $2$cos$^2(z) = 1+$cos$(2z)$ using power series. I know that cos$(z) = \sum (-1)^n\frac{z^{2n}}{(2n)!}$ I also know that if $a(z) = \sum a_nz^n$ and $b(z) = \sum b_nz^n$ then $a(z)b(z) ...
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2answers
187 views

Convergence of a Complex Power Series at the radius of convergence

I am currently reviewing some complex analysis, and have come across this question which I absolutely have no idea on how to attempt: Suppose the radius of convergence of the power series $f(z) = ...
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2answers
228 views

Geometric Series with coin tosses

Suppose you toss a coin and observe the sequence of $H$’s and $T$’s. Let $N$ denote the number of tosses until you see “$TH$” for the first time. For example, for the sequence $HTTTTHHTHT$, we needed ...
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33 views

how to prove uniform convergence of truncated product $\cos_n(z)$ to $\cos(z)$ in the strip $\Im(z)<1$?

The function $\cos(x)$ can be expressed as an infinite product in terms of its zeros $$\cos(z)=\prod_{k=0}^{\infty}\left(1-\frac{z^2}{((2k+1)\pi/2)^2}\right)\tag{1}$$ Let us define ...
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2answers
50 views

How do you show that a power series of $e^x$ at a point converges to $e^x$?

My task is this: Find Taylor-expansion of $e^x$ at the point $1$, the convergence interval $I$ and then show that the series converges to $e^x$. My work so far: By using $Te^x @ x= 1$ we should get ...
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2answers
22 views

Designing a Power Series with certain $R$

Out of interest, is there a way to design a series with a certain radius of convergence? For example, $R=8$, or is there a way to turn a series for which the Radius of Convergence is known, then ...
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1answer
47 views

Writing the product $\sum\limits_{r=0}^\infty \frac{z^r}{r!} \sum\limits_{s=0}^\infty \frac{z^{-s}}{s!}$ as a power series in $z$

My lecturer states that the product $$\sum_{r=0}^\infty \frac{z^r}{r!} \sum_{s=0}^\infty \frac{z^{-s}}{s!}$$ can be written as (with $n = r-s$) $$\sum_{n=0}^\infty z^n\sum_{r=n}^\infty ...
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1answer
8 views

Radius of convergence of complex power series using Cauchy's integral formula

I have a question as follows. Let $$f(z)=\frac{\sin z}{(z-1-i)^2}$$ and $$a_n=\frac{f^{(n)}(0)}{n!}$$ Determine the radius of convergence of $$\sum_{n=0}^{\infty}a_nz^n$$ In my class we have ...
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1answer
37 views

$n$th root of power series when its coefficients are from a field with positive characteristic

Let $k$ be algebraically closed field of characteristic $p>0$. Let's consider a power series $f(x,y)\in k[[x,y]]$. Under what conditions (on $n$, $f$, ...) there exists $g(x,y)\in k[[x,y]]$ such ...
2
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1answer
39 views

Radius of convergence from recurrence with variable coefficients

I am solving via power series the ivp $$y'-2xy=0,\quad y(1)=2.$$ The "solution" is $$y(x)=2\left(1+2(x-1)+3(x-1)^2+\frac{10}{3}(x-1)^3+\frac{19}{6}(x-1)^4+\frac{26}{10}(x-1)^5+\cdots\right)$$ with ...
0
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1answer
696 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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0answers
24 views

Calculate the radius of convergence of the following power series

Let the power serie $\sum_{k\ge0}a_k(z-a)^k$ have the radius of convergence $\rho=t\in\mathbb{R^+}$, and let $p\in\mathbb{N}$. What is the radius of the following series: a) ...
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2answers
41 views

To simplify the series of matrices

Let $A$ be a square matrix in the form $A=B+O(h^2)$, where $B$ is a fixed matrix, and $O(h^2)$ is a matrix with very small elements. Assume that: $$(I-A)^{-1}=I+A+A^2+A^3+...$$ How can I esimate the ...
2
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1answer
53 views

Does $\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n$ converge at the endpoints of the convergence radius?

My task is this: Find the convergence radius of$$\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n.$$ My work so far: By ratio test we get ...
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1answer
32 views

Frobenius Method recurrence relations

Q: By seeking a power series solution to $$2xy′′+(3−x)y′−y = 0$$ about $x=0$ show that there are two linearly independent solutions that have the recurrence relations $$a_{n+1} =\frac{a_n}{2n+3}$$ ...
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2answers
44 views

Find the fourth Taylor polynomial of f(x)=ln(x+1) at x=1

Let $f(x)=\ln(x+1)$ then (a) find the fourth Taylor polynomial of f at x=1 and (b) use part (a) find the approximate the value of ln(2.2) correct 4 decimal (c) Find an estimate for the error in ...
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2answers
33 views

Let $s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$, $c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$ prove that: $c(x)^2-s(x)^2=1$

Let $$s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$$ $$c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$$ prove that $$c(x)^2-s(x)^2=1$$ I know that the following series are representations of the cosh ...
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1answer
37 views

Does this matrix series have an answer?

I'm trying to solve this series: $$\displaystyle\sum_{i=0}^{k}A^i B C^{k-i}$$ Where A, B, and C are $N\times N$ symmetric matrices. And $A$ and $C$ have spectral radii smaller than or equal to 1, ...
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2answers
52 views

Condition for convergence of infinite sum $\sum_{k=1}^{\infty}\frac{x^k}{k} $

Consider the following: $Q= \displaystyle\sum_{k=1}^{\infty}\frac{x^k}{k} $. What condition is required for $x$ so that $Q$ becomes convergent?
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2answers
39 views

Application of power series/ binomial theorem in inverse sampling

I have posted this already in other forums. Apologies for cross posting. In order to establish some properties of inverse sampling, Haldane (1945) uses power series and the binomial theorem I ...
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1answer
42 views

Interchanging limit and double series

I have a generating function $$U(z,w)=\sum_{j=0}^{\infty}\sum_{n=0}^{\infty}u_{j,n}z^jw^n$$ Where $0<z<1$, $0<w<1$, $0\leq u_{j,n}<1$. Is is true for this series that $$\lim_{z \to ...
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0answers
13 views

Hadamard's theorem; redefining indexing variable

I have seen in a few proofs the use of Hadamard's theorem to prove convergence of series like this: $\sum_{n\geq 0}z^{n!}$, or $\sum_{n\geq 0}z^{n^2}$ through simply changing the variable of indexing ...
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1answer
17 views

Annular regions in which the Laurent series converges

For the series $$\sum^\infty_{-\infty}\frac{z^n}{3^n + 1}$$ Determine the annular region in which this series converges. I understand that $\sum^\infty_{-\infty}\frac{z^n}{3^n + 1}$ can be split into ...
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1answer
61 views

Find the value of $\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}$ [closed]

$$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$$ I found this question during my study. In my opinion ,it is not difficult to solve ,but it is interesting. So I ...
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3answers
33 views

Explanation of the Sum of an Infinite Series Equation

I've been presented with the following infinite sum (where $P$ is the probability of an event, and $1-P$ is therefore the probability of it not occurring. I was given the following equation as fact: ...
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1answer
27 views

Find all the $z \in \Bbb{C}$ such that the following series converges:

Find all the $z \in \Bbb{C}$ such that the following series converges: $$\sum_{n=0}^{\infty}\frac{(z+i)^{3n}}{(n^3 + 1)^{1/3}e^{3n}}$$ To solve this problem I proceed as usually, first of ...
2
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1answer
18 views

Complex variable: studying convergence of series in terms of radius of a different series

Trying to solve this problem: If the radius of convergence of the power series $$\sum_{n=0}^\infty a_n z^n$$ is R, with $0 < R < \infty$, then the radius of convergence $R_1$ of the ...
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1answer
27 views

Finding power series and ROC of complex function

I have the function $f(z) = \frac{3iz-6i}{z-3}$ I need to find a power series $\sum c_n (z-1)^n$ about $z_0 = 1$ I can rewrite $f$ as $\frac{2i-iz}{1-\frac{z}{3}}$, where I'm guessing the ROC ...
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0answers
20 views

Evaluation of a series with absolute value

I want to estimate or evaluate the series $$S(\xi)=\sum_{n=1}^\infty\beta_n\left|\sin(\pi n \xi)\right|,~~ \xi\in(l_0,l_1)$$ with $\beta_n=\frac{\omega\sin\left(\pi^2 n^2 ...
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1answer
65 views

What is this infinite summation?

We encountered an function defined by the infinite summation as shown below: $$F(x,a):=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}2^{-n}(2a-n)x^{n-1}\Gamma(2a+1)}{a(2a-1)\Gamma(2a+1-n)}$$ Where ...
1
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3answers
60 views

Calculate $\sum_{n=0}^\infty(n+2)x^n$

I am trying to calculate $\sum_{n=0}^\infty(n+2)x^n$. I was thinking it is like the second derivative of $x^{n+2}/(n+1)$ but I am not sure how to go about calculating it. Any hints?
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1answer
15 views

Relation between coefficients of two different power series.

Let $$f(z) = \sum_{n\geq 0} = a_nz^n, a_n\in\Bbb{C}$$ has a radius of convergence $\rho$. Then we can write $f(z) = \sum_{n\geq 0} b_n (z-\frac{\rho}{2})^n$ for $\{z: ...
10
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2answers
372 views

How can we show that $ \sum_{n=0}^{\infty}\frac{2^nn[n(\pi^3+1)+\pi^2](n^2+n-1)}{(2n+1)(2n+3){2n \choose n}}=1+\pi+\pi^2+\pi^3+\pi^4 ?$

We proposed this sum, but we are lacking in knowledge of this area of maths and we would ask if any of the authors would be willing to show us step by step how to go about proving this sum. $$ ...
0
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0answers
43 views

Compute the radius of convergence and interval of convergence of $\sum_{n=1}^\infty\left (\frac {4+2(-1)^n} 5\right)^nx^n $

Compute the radius and interval of convergence of: $$\sum_{n=1}^\infty \left(\frac {4+2(-1)^n} 5\right)^nx^n .$$ I went about this question by applying the root test and this is what I have gotten ...
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1answer
23 views

Mclaurin series and n-th derivative

(1) Find the general formula of the McLaurin series of $ f(x) = arctan((x^3)/2)/x^3\ $ (2) Evaluate the 18-th derivative of f(x) (3) Evaluate lim to infinity of f(x) By general formula do we just ...
1
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2answers
45 views

Power Series Expansion $e^{2x}$

I'm currently finding the power series expansion centered at 0 for a bunch of functions. My answers are starting not to add up with the 'correct' ones and I'm looking for some assistance as to what ...
0
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1answer
30 views

Series Solution to an ODE

I'm trying to complete a practice exam for a class, but I'm having problems verifying my answers. I looked online, but I couldn't find any applet that calculate power series solutions to equations ...