# Tagged Questions

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### Why does these complex sequences converge uniformly?

I have one complex series and one sequence. It is used in complex analysis in a part of my book where they are integrated. However, as you know in order to change limit and integration order it has to ...
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### How to show that $\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$

Let $x_n$ be a sequence of reals. Show that $$\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$$ Since the weak convergence is equivalent to pointwise convergence of characteristic functions ...
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### Absolute convergence of series $\sum_{n=1}^\infty \frac{1}{\frac{1}{z^n}-1}$

\begin{align} \sum_{n=1}^\infty \frac{1}{\frac{1}{z^n}-1} \end{align} Determine the values of $z,z\in\mathbb{C}$ so that the series converges absolutely I know that the series converges for ...
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### 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 ...
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### 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 $\frac{z^{(n+1)!}}{z^{n!}}$ ...
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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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### 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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### Prove that $\sum_{n=0}^{\infty}e^{in\theta}$ is bounded

For my homework class, we need to prove that a certain series converges, for which it is useful to use that this series is bounded ($\theta \in (0,2\pi)$): $$\sum_{n=0}^{\infty}e^{in\theta}$$ I ...
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### How to calculate $\displaystyle\lim_{n\to\infty}\sqrt[n]{|1-z^n|}$ with $z\in\mathbb{C}$ and $|z|\ne 1$?

As stated in the title: How does one calculate $$\displaystyle\lim_{n\to\infty}\sqrt[n]{|1-z^n|}$$ with $z\in\mathbb{C}$ and $|z|\ne 1$?
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### Show uniform convergence of a series of complex function on every compact subset

Let $f:B(0,1)\rightarrow \mathbb{C}$ be an analytic function. Suppose $\sum^\infty_{n=0}f^{(n)}(0)$ converges absolutely. Show that there exists an entire function $g(z)$ such that $g(z)=f(z)$ for ...
I have the series $\displaystyle\sum_{n=0}^{\infty} \dfrac{z^{2n}}{(2+i)^n}$, and I need to find its radius of convergence. I cannot think how to arrange this so I can find $R= \lim ... 1answer 28 views ### Problems with proving that a sequence converges to some certain limit So here is my problem, I would like the prove the following, For any$m=0,1,2...$and for$\alpha\in \mathbb C$with$\Re(\alpha)<0\$ it holds that, $$\lim_{t\rightarrow\infty}t^me^{-\alpha ... 2answers 44 views ### Convergence of complex power series question I need some help to solve this problem and find the domain of convergence of the following power series:$$\displaystyle\sum_{n=0}^\infty(2^n+i^n)(z-2i)^n$$Thank you! 3answers 55 views ### Showing that the sequence z^n is normally but not uniformly convergent [duplicate] I was able to show that z^n is normally convergent on the unit disk centred at the origin, but I am not sure how to show that it is not uniformly convergent on the unit disk centred at the origin. ... 1answer 41 views ### Showing that \sum\limits_{n=1}^{\infty}f_n-f_{n+1} converges normally if f_n converges normally A function sequence \{f_n\} converges normally in an open set U \subseteq \mathbb{C} to the limit function f. Show that the series \sum\limits_{n=1}^{\infty}(f_n-f_{n+1}) converges normally ... 1answer 36 views ### Property of a normally convergence squence of functions Let the terms of a function sequence \{f_n\} be continuous in an open set U \subseteq \mathbb{C}. If \{f_n\} converges normally in U to the limit function f and if z_n \to z_0 in U, ... 1answer 84 views ### sequence uniformly convergent on the boundary of a bounded set in \mathbb{C} Let (f_n) be a sequence of functions which are analytic on a bounded region A\subset\mathbb{C} and continuous on the closure Cl(A). Suppose that the sequence is uniformly convergent on the ... 0answers 216 views ### What does the convergence of a Dirichlet series tells us about the convergence of a power series? If D(s)=\displaystyle \sum_{k\geqslant 1} f(k)\, k^{s} converges for \Re(s)\lt a, what is the radius of convergence of \displaystyle \sum_{k\geqslant 1}f(k)\, x^k =T(x)? Conversely, what ... 1answer 25 views ### Finding the Power Series of a Complex fuction. Find a power series expression \sum_{n=0}^\infty A_n z^n  for  \frac{1}{z^2-\sqrt2 z +2}  I'm completely stuck on this question. I know how to manipulate power series but I've never had to find ... 1answer 56 views ### absolute convergence of infinite product We know if \sum_{n=1}^\infty|z_n| converges then \sum_{n=1}^\infty z_n converges absolutely. (kind of trivial) I wonder whether it holds for infinite products, that is, if ... 0answers 28 views ### convergence of two complex series I want to know whether the following statement is true or not. \sum_{n=1}^\infty \log(1+z_n) converges iff \sum_{n=1}^\infty z_n converges. I know \sum_{n=1}^\infty \log(1+z_n) converges ... 1answer 57 views ### Convergence question and degree of polynomial I'm currently teaching myself power series and Taylor's theorem for complex analysis and I'm having trouble answering questions of the following form: 1) Suppose the power series ... 0answers 83 views ### Normal convergence versus uniform convergence I am wondering about the nature of uniform and normal convergence. I know that uniform convergence is a weaker condition than normal convergence and that normal convergence even implies uniform ... 4answers 170 views ### Ratio test: n^\sqrt{n} I need to determine the radius of convergence of:$$\sum_{n=1}^\infty z^n n^\sqrt{n}$$I have, by use of the ratio test, written: (Because I know it tends to 1)$$\lim_{n\to\infty} ...
How can I calculate radius of convergence of the following series? $$\Large \sum\limits_{n=0}^\infty \frac{5^{n+1}}{\sqrt[n]{(2n)!}}z^{n}$$ I tried using D'alembert convergence test but cannot ...