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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2
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1answer
48 views

Simplify $\lim_{n\rightarrow \infty }\frac{((n+1)!)^{k+9}((k+9)(n))!}{((k+9)(n+1))!(n!)^{k+9}}|x|$

Where k is an unknown positive constant. I get to the point where $\lim_{n\rightarrow \infty }\frac{(n+1)^{k+9}((k+9)(n))!}{((k+9)(n+1))!}|x|$ but I'm not sure how I can further simplify or if this is ...
1
vote
1answer
436 views

Radius of convergence of a power serise involving the Fibonacci sequence.

Consider the power series $$\sum_{n=0}^{\infty}a_nz^n.$$ where, $a_0=0$ , $a_1=1$ , $a_n=a_{n-1}+a_{n-2}$. Find the radius of convergence of the power series. MY Attempt : Clearly $\{a_n\}$ is a ...
2
votes
1answer
162 views

Infinitely nested radicals

In a recent paper it was stated (and maybe proved) that we can solve any polynomial equation with nested radicals. Here "nested radicals" means expression such as: $$ ...
3
votes
1answer
38 views

Power series converges to $\{0\}$ or $ \mathbb{R}$

Given this theorem: If a power series $\sum_{n=0}^\infty a_n x^n$ converges at some point $x_0 \in \mathbb{R}$, then it converges absolutely for any $x$ satisfying $|x| < \left|x_0\right|$. ...
1
vote
1answer
49 views

Conditions on coefficients of complex power series to ensure it is real

Given a complex valued function $f(z)=\sum_{n=0}^{\infty} a_nz^n$ with radius of convergence $R>0$, and $\rho\in (0,R)$, is there an if and only if giving that $f([0,\rho])\subset \mathbb{R}$? So ...
3
votes
2answers
80 views

Compute Power Series Convergence to a function

Consider the next power series $$ \sum_{n=1}^{\infty} \ln (n) z^n $$ Find the convergence radius and a the function $f$ to which the series converges. I have easily found that $R=1$ is the ...
0
votes
1answer
69 views

Does $\sum (2n)!/(n!) $ converge p-adically

Does $\sum (2n)!/(n!) $ converge p-adically, I have worked out $v_p((2n)!) \leqslant 2n/(p-1) $ similarly $v_p((n)!) \leqslant n/(p-1) $ I want to prove this using the result that it converges ...
1
vote
1answer
74 views

Radius of Convergence of Power Series $\sum_{n=0}^\infty\frac{\tanh^{(n)}(0)}{n!} z^n$

What is the radius of the power series $\sum_{n=0}^\infty\frac{\tanh^{(n)}(0)}{n!} z^n$? Justify your answer. My steps toward a solution I can express $\tanh$ simpler as: \begin{align*} \tanh z ...
4
votes
2answers
81 views

Convergence of $\sum \frac{(-1)^{n+1}}{n}z^n$ at $|z|=1$

I know that the power series $\sum \frac{(-1)^{n+1}}{n}z^n$ converges for $|z| \lt 1$ but I have been trying to determine what happens on $|z|=1$ Clearly the series converges at $z=1$ and diverges at ...
0
votes
2answers
49 views

Why is the radius convergence of $\sum_{n=0}^\infty \frac{x^{4n+1}}{4n+1}$ is $1$?

Why is the radius convergence of $\sum_{n=0}^\infty \frac{x^{4n+1}}{4n+1}$ is $1$? We know that $$\frac{1}{R} = \limsup_{n\to\infty} \sqrt[n] {\frac{1}{4n+1}} = 0$$ And therefore, $R=\infty$. ...
0
votes
2answers
171 views

To the power of n/2 - how to get rid of it.

In short, I have a formula $ \frac{2}{3}((-2)^{\frac{n}{2}}-1) $ I need to get rid of (n/2). I have to make sure that I only raise the number with degree n, the integer part, not n/2. Any ideas?
0
votes
3answers
34 views

Finding R for a power series

Let $\sum_2^\infty a_nx^n$ be a power series. Find the radius of convergence when $\lim \limits_{n \to \infty} \frac {a_n}{n^3}$ = 1. I've tried using root test but that gets messy, can't find a way ...
1
vote
1answer
66 views

Radius of convergence of a power series

Let $\sum_2^\infty a_nx^n$ be a power series. Find the radius of convergence when $\lim \limits_{n \to \infty} \frac {a_n}{3^n}$ = 1. There are a few more questions in this manner, but I'd like to ...
0
votes
2answers
62 views

Show divergence of this innocent series

Let $a$ be a complex constant with norm $|a| > 1$. Show that the series $$\sum_{k=1}^\infty \frac{a^k}{k^4}$$ diverges. The problem is really easy if $a \in \mathbb R$ is real: one just have to ...
1
vote
2answers
80 views

Find $\lim\limits_{n \to \infty} \frac{\log(1+2^n)}{\log(1+3^n)}$

How to calculate this limit? $$\lim\limits_{n \to \infty} \frac{\log(1+2^n)}{\log(1+3^n)}$$
2
votes
2answers
41 views

Power series area of convergence with $\sin$

I want to examine for which $x\in\mathbb{R}$ the series $$\sum_{n=1}^{\infty}(2x-1)^n\sin\left(\frac{1}{n^2}\right)$$ converges. So far I have tried to use the inequality ...
0
votes
0answers
64 views

Radius of convergence $\sum_{n=1}^{\infty} \frac{x^n}{1+x^n} $

I want to prove that the radius of convergence of the series: $$\sum_{n=1}^{\infty} \frac{x^n}{1+x^n} $$ is $r=1$. Yeah in this interval does converge by comparison with the geo series ...
1
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3answers
157 views

$x+2$ is irreducible in the power series ring $\mathbb{Z}[[x]]$

For the last few days I am trying to prove that $x+2$ is irreducible in $\mathbb{Z}[[x]]$. I think that it is false... I would be very much thankful for any kind of suggestions and help.
1
vote
1answer
59 views

What function does this series represent?

A midterm I proctored recently showed that $$ \cos(\sqrt{x}) = \sum_{k=0}^{\infty} \dfrac{(-1)^k x^k}{(2k)!}$$ The question asked what function this series represents. It may represent cosine, but ...
0
votes
1answer
79 views

Power series divergence (real analysis)

Show that if a power series diverges at $x_0$ then it must also diverge when $\lvert x\rvert > \lvert x_0\rvert$ or provide a counterexample. I feel like there is a counterexample for some kind of ...
3
votes
1answer
22 views

problem convergent power series expansion such that $f^{(n)}(x)$ and $|f^{(n)}(x)|\leq 1$ for every $n\geq 1$ and for every $x\in(-1,1)$

Let $f:(-1,1)$ $\to \mathbb{R}$ such that $f^{(n)}(x)$ exists and $|f^{(n)}(x)|\leq 1$ for every $n\geq 1$ and for every $x\in(-1,1)$. Then f has a convergent power series expansion in a neighbourhood ...
1
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2answers
66 views

Non-trivial examples of power series which are uniformly convergent on $[0,1)$ and left-continuous at $x = 1$

The question is motivated by a more extensive problem that needs a formal proof, but I am not interested in help on the proof itself, but I'd like to see some examples of such power series. I put ...
4
votes
2answers
93 views

Prove $\lim\limits_{n \to \infty} \sup \left ( \frac{(2n - 1)^{2n - 1}}{2^{2n} (2n)!)} \right ) ^ {\frac 1 n} = \frac {e^2} 4$

This is a problem in Heuer (2009) "Lerbuch der Analysis Teil 1" on page 366. I assume that the proof should use $e = \sum\limits_{k = 0}^{\infty} \frac 1 {k!}$, but I cannot come further.
0
votes
1answer
52 views

finding the radius of convergence of a complex power series

I am trying to find radius of convergence of $$ \sum_{n=0}^{\infty} z^{a^n} $$ where $a>1$ integer. I obviously want to use $1/R = \limsup ( |c_n| )^{1/n}$. Is there a way to write $z^{a^n}$ ...
1
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0answers
22 views

Divergent Sum Renormalisation

I noticed an interesting property of holomorphic functions and I'm wondering if it forms the basis of divergent sum renormalisation. Let $f,g:\mathbb C \rightarrow \mathbb C$ be holomorphic ...
1
vote
1answer
359 views

Complex Arctan function and its power series

I face a sequence of confusing questions: In complex plane, note that $arctan(z)$ denote the principal branch of inverse complex tanget function ,by requiring $$\frac{-\pi}{2} < ...
3
votes
1answer
100 views

Is this correct reasoning about Taylor series?

Is the following correct reasoning about the Taylor series? I'm just trying to build some intuition but just want to make sure it's correct. If a function $f(x)$ has a power series representation ...
0
votes
3answers
90 views

Power Series (Laurent Series)

I need some help with this exercise: I need to obtain the power series development of this function: $$f(z)=\frac{\cos(z+1)}{(z^2-1)z}$$ Centered in $z_0=-1$ and valid in $z_1=\frac{1}{2}-i$ I know ...
1
vote
1answer
72 views

Help: infinite sum for matrices

Suppose $G$ is an $n\times n$ matrix. Can someone show me how you can carry $$I + G x^{-1} + G^2 x^{-2} + G^3 x^{-3} + G ^4 x^{-4} + \cdots$$ to $$(xI - G)^{-1}x$$ without having to "divide" ...
1
vote
2answers
100 views

Prove that $f$ has derivatives of all orders at $x=0$ [duplicate]

Let $\displaystyle f(x) = \begin{cases}e^{- \frac{1}{x^2}} &\text{for } x \neq 0 \\ 0 & \text{when } x=0 \end{cases}.$ Prove that $f$ has derivatives of all orders at $x=0$, and ...
0
votes
1answer
56 views

Maclaurin Series with Power in Denominator?

$$f(x) = \frac {x}{({125+6x^2})^{1/3}} $$ I'm having a bit of a tough time trying to figure out this question (in which I'm supposed to find the first five coefficients after creating a Maclaurin ...
1
vote
3answers
77 views

How to prove $\sum_{k=0}^{\infty}k^2x^{k} = \frac{x(1+x)}{(1-x)^3}\text{, }|x| < 1$? [duplicate]

How do I prove that the summation $$\sum_{k=0}^{\infty}k^2x^{k} = \dfrac{x(1+x)}{(1-x)^3}\text{, }|x| < 1\text{?}$$
2
votes
1answer
48 views

For every $z\in \Bbb C$, the exponetial series converges uniformly on every bounded subset of the complex plane

$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$ This series converges uniformly on every bounded subset of the complex plane. What does this mean in simple terms?
1
vote
1answer
90 views

Exponential of a complex number converges absolutely

$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$ This converges absolutely for every $z\in \Bbb C$. What does this mean to a layman?
4
votes
1answer
83 views

Series (Dilogarithm Function)

Let $\displaystyle f(x)=\sum_{n=1}^{\infty} \dfrac{x^n}{n^2} , \; x \in (0, 1)$. Evaluate $f(1/2)$ without using the known formulae of the dilogarithm or the equation it satisfies. May I have some ...
1
vote
1answer
48 views

The radius of convergence of a power series about a point interior to the domain of an analytic function

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a real analytic function with domain an open, non-empty set $(a, b) \subseteq \mathbb{R}$, $-\infty \leq a < b \leq \infty$ and let $c \in (a, b)$. ...
0
votes
2answers
31 views

sequence power series simplification

Let $\{a_n\}$ be the sequence $a_n=\sqrt5\left(\frac{3+\sqrt5}2\right)^n - \sqrt5\left(\frac{3-\sqrt5}2\right)^n$ for each $n\ge 0$. Determine a rational expression for C(x) = $\sum_{n\ge0}a_nx^n$ ...
4
votes
0answers
111 views

How to solve this equation in $ \mathbb{C} $?

From a small simple calculation , we get the following formulas: $ \begin{cases} e^x = \Big( \displaystyle \sum_{n \geq 0} \frac{x^{3n}}{(3n)!} \Big) + \Big( \displaystyle \sum_{n \geq 0} ...
0
votes
1answer
38 views

isomorphism between $k[[x]]$ into $\varprojlim_n k[x]/(x^n)$ [duplicate]

i want to find isomorphism between $k[[x]]$ and $\varprojlim_n k[x]/(x^n)$ but I cant.please help me to find this.
4
votes
1answer
177 views

Formal power series ring, norm. [closed]

Let $k$ be a field. Let $R$ be the formal power series ring $k[[x]]$. Define $N$ on $R \setminus \{0\}$ as follows: $N(f)$ is the smallest $n$ of which the coefficient of $x^n$ in $f$ is nonzero. (a) ...
3
votes
0answers
41 views

Power Series : Interval of Convergence

Find the interval $I$ and radius of convergence $R$ for the given power series. $$\sum_{n=1}^\infty \frac {5^n}{n}x^{n}$$ What I got was that I used the limit as it goes to infinity I ended up with ...
2
votes
1answer
176 views

If $f$ is analytic, prove that $\overline{f(\overline{z})}$ is also analytic

Let $f$ be an analytic function in an open set $U \subseteq \mathbb{C}$. Let $V=\{z\in\mathbb C:\overline z\in U\}$. Define $g$ on $V$ by $g(z)=\overline{f(\overline{z})}$. Show that $g$ is analytic ...
0
votes
1answer
28 views

Expand rational fractional expression in power series $\frac{k^2}{(k^2+\frac{1}{2})^{n+2}}$

Expression as showed in the title and n can be take arbitrary natural numbers, 0,1,2,.... How to expand it in powers of k and what coefficient of $k^m$ is? Any suggestion is much appreciated.
0
votes
2answers
27 views

Taylor expansion of at a point different from $0$: should the variable be changed?

Find the Taylor expansion of $\arcsin x$ at point $1$. Can we change variable to get the series at point $0$? If yes how, and when do we change again to get back to $1$? More generally Let's ...
2
votes
3answers
211 views

What does it really mean for the power series of a function to converge?

So I was watching a Khan Academy video about power series https://www.khanacademy.org/math/integral-calculus/sequences_series_approx_calc/power-series-algebra/v/rep-function-with-geometric-series; It ...
1
vote
2answers
163 views

Identifying a function from its power series representation

What functions are represented by the following power series? $$\sum\limits_{k=1}^{\infty}kz^k \quad \quad \quad \sum\limits_{k=1}^{\infty}k^2z^k$$ Would this involve using a Taylor expansion? I ...
3
votes
1answer
94 views

How to evaluate sums like this $\sum_{i=0}^{\infty}\frac {1} {4^{2^{i}}}$

I thought about: -Transform in a product of two sums and use Cauchy product. -Complete this series into the geometric series. The second one seems more likely to work, but I keep counting stuff ...
5
votes
1answer
120 views

Unknown Taylor expansion

I have come across a few apparently related Taylor expansions, as detailed below: \begin{align} &\dots\frac{a^7}{140}-\frac{a^6}{80}-\frac{3 ...
1
vote
1answer
53 views

Exponential of little o

I have a series and the error is of order $o(x^N)$: $f(x)=\sum_1^N x^n+ o(x^N)$ Now I want to take exponential on both sides. What happens with the error part?
0
votes
1answer
18 views

Convergence radius argument

I'm studying complex function theory and I ran into this argument made by my prof but I can't really wrap my head around it. Set $f(z):=\frac{1}{7+z^2}$ Now notice $\sum_{n=0}^\infty ...