1
vote
2answers
146 views

Regarding the radius of convergence and its equality to a certain limit

Let $f$ be a holomorphic function on the open unit disk $\mathbb{D}$, and suppose that $f$ cannot be extended holomorphically to any open set $\Omega$ containing $\overline{\mathbb{D}}$. Let $f(z) = ...
1
vote
2answers
28 views

Find relative radius of convergence for this seies

Given two series $\sum _{n=1}^{ \infty} a_nz^n$ and $\sum _{n=1}^{ \infty} b_nz^n$ who both have radius of convergence $R$, show that the radius of convergence for $\sum _{n=1}^{ \infty} c_nz^n$ is at ...
1
vote
2answers
57 views

Taylor series of Infinitely differentiable function with nonnegative derivatives

Let $f(x)$ be a nonnegative and infinitely differentiable function on $[-a,a]$ to $\mathbb{R}$ such that $\forall x\in[-a,a]:f^{(n)}(x)\ge0$. Prove that the series: $$\sum_{i=1}^\infty ...
4
votes
1answer
107 views

Convergence of the power series $\sum \left(\frac{n^n}{n!} x^n \right)$

Find the convergence radius of the serie $$\sum \frac{n^n}{n!}x^n $$ and analyze the absolute convergence and/or uniform. What I've done: It is easy to show that the radius of convergence of this ...
0
votes
1answer
28 views

Determining the domain of holomorphic function, the taylor series of function with its convergence's radius.

I need some help and correct my knowledge, please. Let $f(z)=(e^{z}-1)/(1+z+z^{2})$. Determine the largest domain $\Omega$ in $\mathbb{C}$ such that $f$ is holomorphic in $\Omega$. Since ...
0
votes
2answers
17 views

Radius convergence of a power series…

"Suppose that $\sum_{n=0}^{+\infty}a_nx^n$ has convergence radius $R$, $R>0, \text{or }R=+\infty$. Proof that the convergence radius of $\sum_{n=0}^{+\infty}na_nx^{n-1}$ is also $R$." This seems ...
1
vote
1answer
20 views

Finding the convergence radius of a power series

Let $\sum\limits_{n=0}^{+\infty}a_nx^n$ be a power series. Prove that If $\large \lim\limits_{n\to\infty}a_ns^n=0,s>0$, then the power series above converges absolutely for $|x|<s$. ...
2
votes
1answer
47 views

Convergence of a crazy power series

"Let $\alpha$ be a given real number, $\alpha>0$ and $\alpha \notin \Bbb{N}$. Proof that the series $$\sum_{n=1}^{+\infty}\frac{\alpha(\alpha-1)(\alpha-2)\dots(\alpha-(n-1))}{n!}x^n$$ converges for ...
1
vote
2answers
44 views

Conditions for convergence of $\sum\limits_{n=1}^\infty{a^nf(n)}$

assume $a>0$, and for all $n$ we have $0 \leq f(n) \leq 1$. Is there a necessary and sufficient condition on the series $f(n)$ for which $\sum\limits_{n=1}^\infty{a^nf(n)}<\infty$ ? Thanks!
0
votes
1answer
14 views

Finding Convergeance sum for two power-series.

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
2
votes
3answers
55 views

What's the radius of convergence for power series of $1/(1-(x+x^2))$? Is it symmetrical?

When reading the section on composition of power series in Book Calculus With Analytic Geometry (George F. Simmons) (2nd edition, pp. 517), the author claimed you can replace $x$ in a power series ...
0
votes
1answer
17 views

Showing that two infinite series converge to the same value

I was preparing for my exam and came across this problem. Show that The series on the left hand side is the power series of $\ln(1+x)$ evaluated at $x=1$. This is what i've done so far. From ...
1
vote
1answer
35 views

Radius of convergence of powerseries containing $(\log n)^n$

$$ \begin{align} \sum_{n=2}^\infty (\log n)^n(z+1)^{n^2} \end{align} $$ What is the radius of convergence of this power-series? I tried applying the root test and the ratio test , but I couldn't ...
0
votes
2answers
83 views

Disk of convergence of the series $ \sum\limits_{n=1}^\infty n!\,(z-i)^{n!} $

$$ \sum_{n=1}^\infty n!(z-i)^{n!} $$ Find the disk of convergence of this powerseries. Can I set $n!=k$ and then deal with $\sum_{n=1}^\infty k z^k$ . On another note $\frac{z^{(n+1)!}}{z^{n!}}$ ...
1
vote
1answer
23 views

Maclaurin series - Approximation and interval of convergence

This is a problem which I should apparently be solving with Maclaurin series, but I failed to do so. So I attempted it with binomial series, with 5 terms and an error less than the requirement in ...
0
votes
1answer
20 views

Power series function - convergence interval

Could someone help me finding the function and convergence interval for following power series? I don't need a step by step answer, but I'm not entirely sure where to start. $\sum_{n=0}^{+\infty} ...
0
votes
1answer
8 views

Sequence Interval of convergence

I could someone help me with the following sequence of functions of which I attempted to find the interval of convergence, but I couldn't get it to match with the solution I get from WolframAlpha ...
0
votes
1answer
48 views

$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$

$$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$$ I found this question during my study. In my opinion ,it is not difficult to solve ,but it is interesting. So I ...
3
votes
2answers
73 views

Radius of convergence of power series which has factorial term

I am trying to find radius of convergence of the following power series: $\sum_{n\geq 1} n^n z^{n!}$ I tried ratio test but it became complicated, I have never seen such radius of convergence problem ...
2
votes
2answers
62 views

Find the first 5 coefficients of the series $\frac{6x}{x+9} = \sum_{n=0}^\infty C_n x^n$

I rewrote the equation series as $$ \frac69 \sum_{n=0}^\infty \left(\frac{-1}{9}\right)^n x^{n+1} $$ And therefore have coefficients of $C_0 = 6/9, C_1 = \left( 6/9 \right) ...
1
vote
4answers
66 views

Why does $\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$ converge conditionally?

Why does it converge conditionally? $$\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$$
1
vote
1answer
40 views

An example of a complex power series. [closed]

I am looking for a complex power series which is convergent for some $z\in\Bbb{C}$ but not absolutely convergent. In other words, $a_0+a_1z+a_2z+\dots$ is convergent but ...
1
vote
0answers
45 views

Which (approximative) methods are there to compute the inverse of a complicated function?

I have a complicated function $f(x)$ for which I want to compute the inverse $f^{-1}$ over a certain range $R(f): a \leq f(x) \leq b$. The only way to find the inverse I can think of is power series ...
1
vote
1answer
72 views

Power series and their inverses (radius of convergence of each)

Suppose I have a power series approximation $y$ to an invertible function $f(x)$, and I know that $y$ convergences around $x$ on an interval $(-R,R)$, $R$ being the radius of convergence. How are the ...
1
vote
1answer
44 views

Finding radius of convergence using root test

Find the radius of convergence of the following power series $$\sum_{n=1}^{\infty} \frac{2^n + 1}{n} x^n.$$ Using the ratio test, I have found that the radius of convergence is $R = \frac{1}{2}$. I ...
1
vote
3answers
88 views

Power Series Convergence comparison

Given $\sum_{n=0}^\infty c_n4^n$ is convergent, can this be used to find the convergence of $\sum_{n=0}^\infty c_n(-2)^n$?
6
votes
1answer
46 views

Divergence set at radius of convergence

I came up with this question on my own while I was musing around reviewing notes. After unsuccessful Google search (thwarted by a deluge amount of webpages on basic calculus), I decided to ask here. ...
2
votes
2answers
45 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then find whether the above series ...
0
votes
1answer
28 views

$\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor\log_2n\rfloor}}{\lfloor\log_2n\rfloor+1}(x-x_0)^n$ convergence/divergence

I have a problem with determining whether these series are convergent/divergent at the endpoints of their radii of convergence. None of the tests or approaches I know seems to by applicable here... ...
0
votes
0answers
32 views

Convergence of two power series

I just wanted to know, whether my results are correct. I should find the radius of convergency in both cases: $\sum_{n=1}^\infty \frac{z^{2n}}{n^23^{n}}$ with a quotient criterion ...
1
vote
0answers
47 views

Sum of a power series

I have to find the sum of this series $$\sum_{n=1}^{+\infty}\frac{x^{n-1}}{n!}$$ Using integral, I got $$\int\sum_{n=1}^{+\infty}\frac{x^{n-1}}{n!}=\sum_{n=1}^{+\infty}\frac{x^{n}}{n\cdot n!}$$ I ...
5
votes
4answers
110 views

Power series convergence radius

My question is: how do I calculate the radius convergence of a power series when the series is not written like $$\sum a_{n}x^{n}?$$ I have this series: $$\sum\frac{x^{2n+1}}{(-3)^{n}}$$ Can I use the ...
0
votes
1answer
28 views

A basic question about the radius of convergence of infinite power series.

I have a somewhat theoretical question to the definition of the radius radius of convergence of infinite power series. According to the definition for a power series $\sum_{n=0}^\infty a_nx^n$ radius ...
0
votes
1answer
43 views

The power series $\sum_0^\infty2^{-n}z^{2n}$ converges, if

The power series $\sum_0^\infty2^{-n}z^{2n}$ converges, if a) |$z$| $\le$ 2. b) |$z$| $<$ 2. c) |$z$| $\le$ $\sqrt{2}$. d) |$z$| $<$ $\sqrt{2}$. Please anyone give me the answer. I think ...
0
votes
2answers
70 views

Prove $f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ defines a continuous function on $\mathbb{R}$.

Prove $$f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k$$ defines a continuous function on $\mathbb{R}$. I think we can show that if $\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ is uniformly ...
0
votes
4answers
67 views

Testing the convergence of a series [closed]

Does the following series converge? $$\sum_{n=0}^{\infty}\frac{1}{n+3}$$ Please explain with any convergence test you used.
0
votes
0answers
48 views

Radius of Convergence and the Frobenius Method

Consider the equation $$4xy'' + 2y'+ y = 0$$ I know that $x=4$ is a regular singular point, and in the notation that my uni uses, we say that: $$(x-x_0)^2 y'' + (x-x_0)p(x)y' + q(x)y = 0$$ where ...
0
votes
2answers
74 views

Radius of Convergence…very tricky question using very little information

Even one of the maths gurus at my university struggled to get a proof out for this... so I'm almost completely lost! This is the question: Let $a_{n}$ be a sequence of positive real numbers for which ...
0
votes
1answer
34 views

Do I have mistakes in my calculations (power series, convergence)?

I'm not sure I got all of these problems right. I'd really appreciate any sort of feedback. For which $x \in \mathbb{R}$ do the following series converge? Problem 1 For ...
2
votes
3answers
22 views

Absolute sequence unbounded within radius of convergence

Let $R$ be the radius of convergence of the complex power series $a_nz^n$ with $0<R<\infty$. Show that if $|z|>R$, then the sequence $|a_nz^n|$ is unbounded. Trying by contradiction: So ...
0
votes
1answer
72 views

Analysis of singularities and taylor representation of $f(z)=\frac{z^2-1}{\sin \pi z}$

Let $$f(z)=\frac{z^2-1}{\sin \pi z}$$ A) Find all singulartities of $f$ in $\mathbb{C}$ and classify each as a pole (specifying the order), essential, removable, or other. B) Explain why $f(z)$ has ...
1
vote
0answers
24 views

What is Radius of Convergence used for?

What is the applications for "Radius of convergence"? I haven't been successful in finding any information about the applications, just a lot of information about how to calculate and what it is... ...
1
vote
2answers
56 views

A question about convergence interval of power series

Could you give me some hint how to solve this problem: Suppose $a_n$ is sequence defined as $a_1=\frac12,a_{n+1}=\frac12\left({a_n}^2+a_n\right)$. I managed to prove that $a_n$ is decreasing ...
0
votes
2answers
82 views

What's the behavior of $\displaystyle\sum_{n=1}^\infty (z+\sqrt{5}+2i)^{n!}$ outside its radius of convergence?

I want to check the behavior of $$\displaystyle\sum_{n=1}^\infty (z+\sqrt{5}+2i)^{n!}$$ outside its radius of convergence. I've tried to use the ratio test as follows: ...
2
votes
1answer
54 views

Series convergence of $\frac{(-1)^n}{x^{2n+1}}$ [closed]

Does this series converge, and if so how would I prove it? I thought of using the ratio test but I'm not sure. The series is $$ \sum_{n=0}^\infty\frac{(-1)^n}{x^{2n+1}}. $$
1
vote
1answer
27 views

$\left(\frac{a_n}{n^k}\right)_n$ is bounded implies $\sum_{n=0}^\infty a_nz^n$ has a radius of convergence $\ge 1$

Let $$\left(\frac{a_n}{n^k}\right)_n\subset\mathbb{C}\;\;\;\;\;(k\in\mathbb{N})$$ be a boundet sequence. I want to show that the power series $$\sum_{n=0}^\infty a_nz^n\;\;\;\;\;(a_n,z\in\mathbb{C})$$ ...
0
votes
1answer
28 views

Convergence of a Power series

Consider the power series $\sum^{\infty}_{n=0} a_nx^n$. It is fairly easy to impose conditions on the value of $x$, so as to make the series convergent. However, I was wondering if it is possible to ...
0
votes
0answers
20 views

Find the radius of convergence of the following

Here I am confused on which method to use, would it be Ratio Test or Hadamards Theorem. Any help would be appreciated.
0
votes
1answer
50 views

Radius of Convergence in Complex Analysis. [closed]

Following Questions are asked in previous years university exams. I'm preparing for the same exam to be held in next month. Please help me to solve these problems. I have no idea how to solve these ...
0
votes
0answers
17 views

If power series converges to 0 $\forall$ $x \in (-R,R)$, then $a_n$ is $0$ for all $n$

Suppose that $$\sum\limits_{n=1}^\infty a_{n}x^{n}$$ converges for $x \in (-R,R)$. Show that if $f(x)=0$ for all $x \in (-R,R)$ then $a_n=0$ for all $n$. When I look at this , my guess is ...