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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0
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3answers
49 views

A simple series

I don't do math a long time, so I completely don't remember how to prove that: $$ \sum_{i=1}^\infty \frac{i}{2^i} = 2 $$ Can anybody help me?
4
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2answers
156 views

Express $(1-z)^{-1}$ as a power series around $z_0=-1+i$.

I need to express $(1-z)^{-1}$ as a power series in powers of $(z+1-i)$. I would like some guidance on the complex analogue of power series and in writing out this particular case. Many thanks for ...
0
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3answers
44 views

How to get power by knowing the number and result

How to get power by knowing the number and result. For Example $$2^n = 8$$ how can i return the power $n$ by knowing number $2$ and result $8$ or $$4^n = 1024$$ how can i return the ...
0
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1answer
56 views

Does $\sum_{n=0}^\infty\frac{a^n}{\frac{n}{2}!}x^n$ converge?

And if so, what is the radius of convergence of $x$? I am inclined to think it converges absolutely for all $x$ but I can't prove it. I have tried using an adaptation of the ratio test: ...
1
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1answer
29 views

A problem about power series and big-O

The problem is: Prove: There exist constants $a$, $b$ such that $\frac{z^3-5z^2+3z}{(z+2)^3}=1+\frac{a}{z}+\frac{b}{z^2}+O(\frac{1}{z^3})$ as $z\rightarrow \infty$ and find an explicit values for $a$ ...
2
votes
1answer
142 views

Formal power series and nilpotent elements

Given a commutative ring $A$, and a finitely presented (associative) $A$-algebra $B$, show that a morphism of $A$-algebras $A[[x]] \longrightarrow B$ is given by evaluation at an nilpotent element $ ...
6
votes
1answer
157 views

The series $2+3x+5x^2+7x^3+11x^4+…$

It occurred to me to ask whether the power series whose coefficients are the primes has non-zero radius of convergence, and if so, what kind of function it produces. Wikipedia has some bounds on the ...
1
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1answer
51 views

Radius of Convergence of $\sum\ z^{n!}$

Does anyone know how to find the radius of convergence of the series $\sum\ z^{n!}$, where $z$ is a complex number? I tried to use the definition: $\frac{1}{R}=Limsup|\frac{a_n+1}{a_n}|$, but I ...
1
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2answers
33 views

Find radius of convergence of power series

Since we know that given $\sum_{n=0}^{\infty }C_nz^n$, if $\lim_{n\rightarrow \infty }|C_n|^{1/n}$ exists then $R^{-1}=\lim_{n\rightarrow \infty }|C_n|^{1/n}$ where $R$ is the radius of convergence. ...
1
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3answers
50 views

Writing an infinite series as the sum of the series

I have $$ y= \sum_{x=1}^\infty \frac{k}{10^{k^{2}}} =0.1002000030000004...$$ I want to be able to write this in terms of a geometric progression. I've tried doing $ 10y = ...
1
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0answers
43 views

Sum of the series problem

I was doing a programming problem.There i faced the difficulty of solving two series. The equation which i was asked to solve was $Z_n+Z_{n-1}+2Z_{n-2}$. Where $Z_{n}$ is given as $P_{n}+S_{n}$. ...
3
votes
1answer
64 views

Showing $|a_k | \le 1$

Let $A$ be the closed unit disk $A= \{z \in \mathbb{C}: |z| \le 1 \}$. Suppose $f$ is an entire function whose Taylor series centered at the origin is $$\sum_{k=0}^{\infty} a_kz^k$$ and that $f$ maps ...
0
votes
2answers
110 views

How to turn $-\ln(1-x^2)$ into a power series representation?

I need to turn $f(x)=-\ln(1-x^2)$ into a power series, there are 2 things I can't understand: 1) I don't how to check and even if to check whether this function can turn into a power series or not, ...
0
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0answers
30 views

Power Series Calculation With Probability Function

Here is the problem I am trying to finish: I have figured out $(a)$ to be $p_n$ because it is part of the summation formula. $(b)$ has to be $0$ because $0^n$ is always $0$, regardless of $n$. I ...
1
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1answer
51 views

Complex Exponential/Trigonometric Functions

I'm having trouble on proving the following state of a Lemma using the power series of $\exp z$ centered at $0$: For all $z \in C$: $\exp(z + 2\pi i) = \exp(z)$ and $\exp(z) \neq 0$ All help ...
1
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2answers
174 views

For which $x$ does the series $∑_{n=1}^∞(1+\frac{1}{2}+ \frac{1}{3}+⋯+\frac{1}{n})\, x^n$ converge?

Determine for what value of $x$ the series converges $$\sum_{n=1}^\infty \left(1+\frac{1}{2}+ \frac{1}{3}+⋯+\frac{1}{n}\right) x^n $$ Observe that $∑_{n=1}^∞(1+\frac{1}{2}+ ...
0
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1answer
52 views

convergence of series, double factorial against power function?

It is known that the series $$\sum_{n=1}^{\infty} \frac{C^n}{n!}<\infty$$ for any $C>0$, that is, the factorial kills the Power function. I wonder now if $$\sum_{n=1}^{\infty} ...
1
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1answer
24 views

Find the closed formula for the number of ways to get n dollars using coins of 1, 2 and 5 dollars

Ok, this is going to be a long one. So, using generating functions I have to find a closed formula for the number of ways to get n dollars if I have infinite amounts of coins of 1, 2 and 5 dollars. ...
3
votes
3answers
71 views

Radius of convergence of the series $\displaystyle\sum\limits_{n=0}^\infty \frac{n!\,z^{2n}}{(1+n^2)^n}$

I am doing the following problem and would like to know whether my answer is correct or not: Find the Radius of convergence for the complex series $\displaystyle\sum\limits_{n=1}^n ...
0
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1answer
26 views

How to find singular point of following diffrential equation?

Here it is the differential equation and whether it is regular or irregular. $ x^2 y’’ + (5/3x+x^2) y’ – y/3 =0 $
0
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1answer
60 views

Power series of $\tan(z)$

In the power series of $$\tan(z)=\sum_{k=0}^{\infty }B_{2k}\frac{(-4)^k(1-4^k)x^{2k-1}}{(2k!)},$$ what is $B_{2k}$? What's the mathematical expression of it? Thanks in advance.
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0answers
40 views

Behaviour of $\sum_{n=1}^{\infty}\frac{z^{n}}{n}$ for $|z| = 1$ [duplicate]

How to show that the power series $$\sum_{n=1}^{\infty}\frac{z^{n}}{n}$$ ( which has radius of convergence $1$ ) converges in all points of $\partial D(0,1)$ except $z = 1$ ?
1
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1answer
55 views

Proof that the series for the generating function of the partition function converges?

For $|q| < 1$, the generating function of the partition function $p(n)$ is given by $$ \sum_{n=0}^\infty p(n) q^n = \prod_{k=1}^\infty {1 \over 1-q^k}. \tag{1} $$ I have an intuitive ...
0
votes
1answer
33 views

How do we prove that all power series are uniformly convegent?

A (complex) power series is always (give or take some extra hypotheses) uniformly convergent on the interior of its disc of convergence. How do we prove this? Also, what is the exact statement? Does ...
0
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2answers
43 views

Proof that the radius of convergence exists

If the radius of convergence is defined as $R$ such that the power series in $x$ (centered at $0$) converges for $|x|<R$ and diverges for $|x|>R$, I would like a proof that this $R$ exists. As ...
0
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2answers
80 views

Trouble with determining principal part of function at a pole

In Fischer's $\textit{A Course in Complex Analysis}$ I am encountering some difficulty in explicitly calculating the principal part of a function at a pole. The function is $f(z)= \frac{1}{z - \sin ...
1
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0answers
35 views

Are there proper infinite power series in $\mathbb Z\to \mathbb Z$?

Are there infinite power series $f:\mathbb Z\to \mathbb Z, \hspace{.5cm} f(n)=\sum_{k\in S\subset \mathbb Z} a_k n^k,$ where $f$ isn't polynomial, i.e. the coefficients $a_k$ (which can be from all ...
0
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1answer
41 views

determine the convergence region of a complex series

Determine the region $\Omega$ of the complex plane such that for any $z\in\Omega$ the following series converges: $\sum_{n=1}^\infty\frac{1}{n^2}\exp(\frac{nz}{z-2})$. I do not know how to treat ...
0
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2answers
24 views

Trouble on proving a lemma - Complex Power Series

I'm having some difficulties on proving the following lemma: "If $f_n$ is a sequence of functions which converges uniformly to $0$ on a set $G$ and $z_n$ is any sequence in $G$ then the sequence ...
2
votes
1answer
35 views

power series for matrix with elements smaller than 1

If I have a square matrix A such that all elements $|a_{ij}| < 1$ does this guarantee that all my eigenvalues will also be less than 1 and that the power series $S = I - A + A^2 - A^3...$ will ...
0
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2answers
30 views

Find the Maclaurin series of $f(x)=\frac{x}{x^4+x^2+1}$ [closed]

Maclaurin series for $f(x)$. Thanks.
0
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3answers
44 views

Problem about ODE and power series

For each $a \in \mathbb{Z}^+$ let the following ODE $$ x'' - \dfrac{a (a+1)}{(1 +t^2)} x = 0$$ Using power series around the origin, show that the equation has a solution $p_a(t)$ which is a ...
1
vote
1answer
51 views

Solve ODE using analytic solutions

Let the following ODE: $x'' + tx' + x = 0.$ Find the general solution $x(t) = a_0 x_1(t) + a_1 x_2(t),$ with $a_0, a_1 \in \mathbb{R}$ and $x_1(t), x_2(t)$ are $t$ power series convergent for ...
1
vote
1answer
81 views

Analytic continuation of complex square root

This example comes from the book by Elias Wegert: Visual Complex Functions. Consider the function $f(z):=z^{1/2}$ for $z \in \mathbb{C}$ with $|z - 1| < 1.$ For these $z$, the function can be ...
1
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0answers
33 views

Find the indicial equation of $(x+2)^2(x-1)y''+5(x-1)y'-\pi(x+2)y = 0$

Find all singular points of each equation, and determine whether they are regular or irregular. At each regular singular point, find the indicial equation and the exponents of singularity. ...
2
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3answers
79 views

Find the form of a second linear independent solution when the two roots of indicial equation are different by a integer

Consider the differential equation $$x^2y''+3(x-x^2)y'-3y=0$$ $(a)$ Find the recurrence equation and first three nonzero terms of the series solution in powers of $$ corresponding to the larger root ...
2
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1answer
44 views

Find one series solution for $xy'' - y = 0$

I have found the recurrence relation to be $a_{n+1} = \frac{(a_n)}{(n+1)(n)}$ . I am stuck at this part because no matter what I set the initial a to be, the following term will have a problem due to ...
0
votes
1answer
29 views

$\lim_{n\rightarrow +\infty} \frac{a_{n}}{a_{n+1}} = z_{0}$ with $z_{0}$ pole [duplicate]

This is an exercise from Stein-Shakarchi. Suppose that $f$ is holomorphic in an open set containing the closed unit disc, except for a pole at $z_{0}$ on the unit circle. Show that if $f(z) = ...
0
votes
1answer
54 views

Integrating a Taylor series term-by-term

Why is $$\int_{0}^{z} \frac{\sin x}{x} \ dx =\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)!} \int_{0}^{z} x^{2n} \ dx$$ not valid for $z= \infty$? Well, at least I'm assuming it's not valid since ...
0
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2answers
28 views

Calculation of a power series sum

How can I calculate the following sum: $$\sum_{n=1}^\infty (n+2)x^n$$ What is wrong with spreading it to: $2x^n + nx^n$? both I know how to calculate. Thank you
1
vote
2answers
84 views

Sum of a power series $n x^n$ [duplicate]

I would like to know: How come that $$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$ Why isn't it infinity?
0
votes
1answer
32 views

Power series (representation) of given function

Well I'm wondering if below power series is the correct result of the function - wolfram alpha doesn't give anything like the result. The function $$\frac{x}{2x^2+1} = x\cdot \frac{1}{1- \left ( ...
0
votes
1answer
40 views

Power series convergence question

Does there exist a sequence $c_{n}$ of complex numbers such that $$ \sum_{n=0}^{\infty} c_{n} z^{n} $$ has radius of convergence $R = \infty$, but for all other sequences $c_{n}'$ of complex numbers ...
0
votes
1answer
25 views

Series expansion of quotients

I'd like to start of with a simple formula from a textbook $T(W) = \frac{W^5}{1-2W} = W^5 + 2W^6 + 4 W^7 + \dots + 2^j W^{j+5} + \dots$ Obviously, this is an expansion of the quotient into a power ...
1
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1answer
26 views

Expanding One Function in Powers of Another

One sees here that it is possible to expand $f(x) = 2x^3 + 7x^2 + x - 6$ in powers of $x - 2$ by taylor expanding $f(x) = f(x - 2 + 2) = f(2 + h)$ about $2$, and this idea can be used in deriving the ...
5
votes
1answer
55 views

Is $\cos x$ irreducible as a power series?

Let $\mathbb{Q}_{\mathrm{ent}}[[x]]$ be the ring of entire functions with rational coefficients. Is $$ \cos x \;=\; \sum_{n=0}^\infty (-1)^n\!\frac{x^{2n}}{(2n)!} $$ irreducible in ...
2
votes
2answers
49 views

Rate of convergence of binomial series

This is the binomial series: $$(1+x)^\alpha = \sum_{k=0}^\infty \binom{\alpha}{k} x^k$$ where $|x|<1$ and $\alpha$ can be a complex number in general. How fast does it converge? I need an upper ...
2
votes
1answer
49 views

Clever way to expand 1/(z^2-n^2) in power series?

Is there a good trick to prove the following identity? $$\frac 1 {z^2-n^2} = -\sum_{i=0}^\infty \frac {z^{2i}} {n^{2(i+1)}}$$ I tried writing out the coefficients as a Taylor series, but this was ...
0
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0answers
28 views

$\|1-f\| < 1 \Rightarrow \exp(\log(f))=f$ if $f$ is in a Banach Algebra.

I'd like to know if it's possible to prove the statement without passing by theorems about power series and convergence radius. The series are as usual: $$\exp(x) = \sum_{n=0}^\infty ...
1
vote
2answers
64 views

$L^p$ spaces and counting measure

currently I am working on the following two exercises as a revision for my exam. Let $\mu$ be the counting measure on $\mathbb N$. Show that if $1 \le p < s < \infty$ then $f \in L^p$ implies ...