# Tagged Questions

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### $\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor\log_2n\rfloor}}{\lfloor\log_2n\rfloor+1}(x-x_0)^n$ 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... ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $z\in A?$ ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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... ...
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### 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 ...
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### 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: ...
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### 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 $$\sum_{n=0}^\infty\frac{(-1)^n}{x^{2n+1}}.$$
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### $\left(\frac{a_n}{n^k}\right)_n$ 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 $$\sum_{n=0}^\infty a_nz^n\;\;\;\;\;(a_n,z\in\mathbb{C})$$ ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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, ...
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### 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.
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### 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 ...
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### 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. ...
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### 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 ...
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### 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!
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### 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 ...
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### 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. ...
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### 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 ...
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### Power series with radius of convergence 2 that diverges at both -2 and 2?

I'm looking for a real power series that has radius of convergence 2 but diverges at both 2 and -2. Any idea? Thank you!
I have to find the radius of convergence of some power series but I find myself in trouble for three of them : the series are $\sum2^kx^{k!}$ $\sum\sinh(k)x^k$ $\sum\sin(k)x^k$. For the first ...
### convergence ratio of the serie $e^{xn}$
How can I determine the values of $x$ such that the series converge: $$\sum_{n=0}^\infty e^{xn}$$ I'm really lost in this problem, please help.