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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3answers
89 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
42 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 ...
1
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1answer
127 views

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

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
30 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
28 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
30 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
90 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 x}e^{\kappa(n)}\left(e^{ia(t)}(\kappa(...
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0answers
33 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 $$f(z)=\...
1
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1answer
32 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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2answers
83 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) = \...
2
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2answers
190 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
232 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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0answers
37 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 $$\cos_n(x)=\...
2
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2answers
51 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
23 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
48 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 \frac{1}{r!(...
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1answer
9 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
41 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
43 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 ...
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0answers
28 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) $\sum_{k\ge0}a_k^m(z-a)^k$...
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2answers
43 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 that$$\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\lim_{n\to\...
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1answer
33 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
48 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
35 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
38 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, i....
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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
40 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 assume....
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1answer
46 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 1-}...
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0answers
16 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
20 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 ...
3
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3answers
46 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 all, ...
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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
28 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
21 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 T\right)}{n^2\left(\omega^2-\...
1
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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 $$a(2a-1)\...
1
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3answers
61 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
17 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: |z-\dfrac{\rho}{2}|<\dfrac{\...
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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
24 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
46 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 I'...
0
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1answer
31 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 ...
0
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1answer
40 views

Taylor series doesn't seem to have a pattern?

My teacher gave us a study guide to work on, and one of the problems doesn't seem to come out right. The directions are to "find the Taylor series of $f(x)=x^5-3x^4+x^3+2x-1$ for $a=1$. I calculated ...
2
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1answer
41 views

What is the radius of convergence of $\sum _{n=0}^{\infty }\left(4+\left(-1\right)^n\right)^nx^n$

$$\sum _{n=0}^{\infty }\left(4+\left(-1\right)^n\right)^nx^n$$ I was asked to find the radius of the convergence, but the Power-Series diverges, so I'm a bit confused. We tried Cauchy-Hadamard and ...
0
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2answers
27 views

Determining radius of convergence by factoring

I know that you can determine radius of convergence of a function $f(x)$ by factoring the function to look like $1/(1-x)$. But when I tried to do that here, it didn't work. This is my work: But the ...
2
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1answer
36 views

Prove that the series $\sum_{1}^{\infty}a^n \frac{logn}{n^2}$ is convergent only for $[-1,1]$ and divergent elsewhere.

Prove that the series $\sum_{1}^{\infty}a^n \frac{logn}{n^2}$ is convergent only for $a \epsilon [-1,1]$and divergent elsewhere. I have given it an honest attempt, I can see why this must be true ...
0
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0answers
79 views

What is the sum of the series $\sum_{n=1}^{\infty}\frac{sin(\frac 1 n)}{n}?$

I want to explicitly calculate the sum of the following series $$\sum_{n=1}^{\infty}\frac{sin(\frac 1 n)}{n}$$ I know that the series converges absolutely because of $|sin(\frac 1 n)|\leq \frac 1 ...
1
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2answers
33 views

How to apply Abel's theorem

In this example, where does the term $log(1+x)$ come from? And can someone briefly summarize the steps we take when we want to apply Abel's theorem? Thanks!