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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1answer
60 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
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1answer
19 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 ...
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2answers
32 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
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2answers
86 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.
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1answer
41 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
17 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
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1answer
92 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
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1answer
67 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 ...
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2answers
53 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
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1answer
49 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" ...
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2answers
43 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
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1answer
34 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 ...
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3answers
51 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
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1answer
36 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
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1answer
23 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
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1answer
62 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
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1answer
39 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
20 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) = $\sum_{n\ge0}a_nx^n$ ...
3
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0answers
66 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
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1answer
34 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.
3
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1answer
38 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
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0answers
25 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
93 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
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1answer
25 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.
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2answers
22 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
75 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 ...
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2answers
43 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
80 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
108 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
31 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
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1answer
16 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 ...
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0answers
25 views

Did i multiply the sums correctly?

This is an extention to this question except i am unsure of whether i have done it correctly: $$ y'' = -y'(f(x) - r(x) y') $$ $f(x) = \sum_{n=0}^\infty s_n x^n$, $y = \sum_{n=0}^\infty a_n x^n$, and ...
0
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2answers
43 views

What is the radius of convergence of $\sum_{k = 0}^{\infty} 3^{k^2}x^{k^2}$

Where I'm at right now: $a_k = 3^{k^2}$, so we do $\frac{1}{\lim sup_{k\to\infty}(3^{k^2})^{\frac{1}{k}}}$, which is $\frac{1}{3^k}$, but that doesn't seem right. I suppose from there, I go through ...
-2
votes
1answer
43 views

Find the interval of convergence for these 3 power series

I believe I need to use the root test and ratio test. I've solved the first two, but I'm not too sure I fully understand how to do these, so was hoping someone else could work them out so I could ...
0
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2answers
43 views

Calculate the radius of convergence of $\sum^\infty_{k=1} \frac{(2k-1)^{2k-1}}{2^{2k}(2k)!}x^k$

I need some assistance on calculating the radius of convergence: $\sum^\infty_{k=1} \frac{(2k-1)^{2k-1}}{2^{2k}(2k)!}x^k$ I tried the quotient criteria: ...
0
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2answers
60 views

Complex power serie : Radius of Convergence

Could anyone please suggest me how to deal with these questions (Complex Variable) : Note that all problems are in $\mathbb{C}$. 3.1 Determine the radius of convergence $\rho$ of each of the ...
3
votes
1answer
53 views

Series approximation to $\int_0^1\sqrt{\frac{2x+3}{2u^3-(2x+3)u^2+2x+1}}du$

I have figured out by graphing that, for small $x$: $$ \int_0^1\sqrt{\frac{2x+3}{2u^3-(2x+3)u^2+2x+1}}du\approx\log(1/x)+\pi/2+O(x) $$ However, I am unable to prove that this is the case. As $x\to 0$ ...
4
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0answers
76 views

Is there a formula for $\sum_{n=0}^{+\infty} q^{n^3}$?

When I studyied the representation of integers as sum of squares, I found that the most powerful tool is the Jacobi Triple Product, in fact this amazing identity allows us to find more useful ...
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0answers
23 views

Using Frobenius to show two solutions

Consider the ODE $$y''+\frac{y'}{x}+\left(-1-\frac{1}{4x^2}\right)y$$ Use the Frobenius method to find a power series solution near $x=0$ corresponding to the smallest root of the indical equation, ...
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1answer
32 views

How to find the Number of factors, if sum of the factors are given?

A number is expressed in terms of $(2^m\times3^n)$, Find the value of $(m,n)$ if sum of all factors of a number is $124$.
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0answers
11 views

The distribution of the random power series

I came across this question below. Let $F(z,w) = \sum_{n=0}^{\infty} X_n(w)Z^n$, where $X_n(w)$ is a random variable, that can be taken either as constant $c_1$ or $c_2$ with same probability of ...
1
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1answer
39 views

What happens to number 3?

I'm reviewing a text on Maclaurin series. This is more of an algebraic question, anyway. How do we go from here: $$ z^2e^{3z} = \sum\limits_{n=0}^\infty \frac{z^2(3z)^n}{n!}$$ to here: $$ ...
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0answers
22 views

Did i generalize this series solution correctly?

$$ f(x) = \frac{y''}{p(x)y'} + r(x) y' $$ if all functions are expressed in their power series form, then: $$ y = \sum_{n=0}^\infty a_nx^n $$ $$ p(x) = \sum_{n=0}^\infty p_n x^n $$ $$ r(x) = ...
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0answers
29 views

Question about multiplying summations with another summation inside

I have the following: $$ y = \sum_{n=0}^\infty [x^n \sum_{k=0}^\infty (k+1)a_{k+1} P_{n-k}] \sum_{n=0}^\infty x^n[s_n - \sum_{k=0}^n a_{k+1}(k+1)R_{n-k}] $$ I can easily multiply $$ ...
0
votes
1answer
37 views

For which $z \in \mathbb c$ does this series converge?

$f(z)=\sum_1^\infty \frac{(2z)^{2k}}{2k(2k-1)}$ I didn't know how to start so I just tried the ratio test. If $|\frac{a_{n+1}}{a_n}|>0$ then the series converges. $\implies$ ...
0
votes
1answer
24 views

Convergency of the power series at two points

Consider the power series $$\sum_{n=0}^{\infty}a_{n}(z+3-i)^{n}.$$ The series converges at $5i$ & diverges at $-3i$. Then which is correct ? (a) convergent at $-2+5i$ & divergent at ...
2
votes
0answers
127 views

How to solve this recurrence relation and solving the power series

Take the following recurrence relation into account: $$ a_{n+2} = \frac{1}{(n+1)(n+2)} \sum_{k=0}^n (s_k - (k+1)a_{k+1})(n-k+1)a_{n-k+1} $$ I know that: $$ s_{2m+1} = \frac{(-1)^m}{(2m+1)!} $$ and ...
2
votes
1answer
23 views

Power series of dependent and independent variables

Let $f(z,w)$ be an analytic function in two variables where $w=w(z)$ is dependent on $z$ ($z$ is the independent variable). Then $f(z,w)$ has a power series expansion centered at $(z_0,w(z_0))$ ...
1
vote
0answers
25 views

Proof that $\sum_{n} a_{n}$ converges if $a_{n}=O(1/n)$, $\lim_{x\uparrow 1}\sum_{n}a_{n}x^{n}$ exists?

Does anyone know of a simple proof that $\sum_{n=0}^{\infty}a_{n}$ converges whenever the real sequence $\{ a_{n} \}_{n=0}^{\infty}$ satisfies these two conditions? $a_{n}=O(1/n)$; $\lim_{x\uparrow ...
0
votes
1answer
19 views

Find the ratio and interval of convergence for $\sum_{n=1}^{\infty} \frac{n!x^n}{1\cdot 3\cdot 5\cdots (2n-1)}$

I believe this would diverge for $x\neq 0$. After using the ratio test I obtain (x)(n+1)(sum from 1 to n of (2n-1)/(2n+1)). Taking the limit as n goes to infinity the second term blows up and the ...