0
votes
1answer
13 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
23 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
39 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
21 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
19 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
7 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
66 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
59 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
61 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
35 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
43 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
62 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
42 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
86 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
42 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
41 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
98 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
27 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
41 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
68 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 ...
3
votes
0answers
70 views

Series that diverges at infinitely many points on the unit circle

Initial problem Given $A=\{\alpha_1,\dots,\alpha_k\}$ with $|\alpha_i|=1$, does there exist a power series $\sum a_nz^n$ that converges everywhere on the unit circle except when $z\in A?$ ...
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
29 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
70 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
33 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
20 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
69 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
77 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
23 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
27 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
19 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
43 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 ...
2
votes
1answer
28 views

Radius of Convergence of Sum of two Series.

Hi all, I know there are similar questions on here, but none deal with the fact of trying to prove that $T \geq min\{R,S\}$. Intuitively this doesn't make sense to me, If you have, ...
1
vote
1answer
81 views

Does the series $\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ converge?

$\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ Please let me know how you did it. Thank you.
1
vote
1answer
52 views

Determine the value of r where the series converges

show that $$ \big(r\big)^{ln(n)} = \big(n\big)^{ln(r)} $$ Then determine the values of r (with r>0) for which the series $$ \sum_1^\infty (\big(r\big)^{ln(n)})$$ converges. r must be in what ...
0
votes
1answer
33 views

Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
0
votes
3answers
67 views

Find a radius of convergence of power series

I have to Find a radius of convergence of this power series I' ve decided to use D'alambert indication: Looking for a limit i meet a problem with a factorial Please. help me finish this ...
0
votes
2answers
41 views

Convergence of complex power series question

I need some help to solve this problem and find the domain of convergence of the following power series: $$\displaystyle\sum_{n=0}^\infty(2^n+i^n)(z-2i)^n$$ Thank you!
1
vote
2answers
69 views

Power Series Representation…

I am having a hard time understanding how to proceed with this question... Find a power series representation for the function and determine it's radius of convergence $$ f(x)= x^2\ln(1+x^2) $$ How ...
1
vote
2answers
58 views

Simplifying ratio test with exponents $k+1$

Question: Find the interval and radius of convergence. $$\sum_{k=1}^\infty\frac{(x-1)^k(k^k)}{(k+1)^k} .$$ I applied the ratio test. ...
1
vote
1answer
25 views

Converting from radius of convergence to interval of convergence

Using the root test I have determined that $$\sum n^{-n} x^n$$ has a radius of convergence of infinity and $$\sum n^{n} x^n$$ has a radius of convergence of 0. Does this mean that the respective ...
1
vote
1answer
106 views

Showing the power series of $\cos(x)$ converges uniformly to $\cos(x)$ on every bounded interval

Show that the power series of $\cos(x)$ converges uniformly to $\cos(x)$ on every bounded interval. My attempt: The power series for $\cos(x)$ is $$\sum_{n=0}^{\infty} ...