All Questions
Tagged with complex-analysis convergence-divergence
842
questions
4
votes
1
answer
178
views
Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.
The standard branch of logarithm, $\log:\mathbb{C}\setminus (-\infty,0]\to\mathbb{C}$ is defined as
$$
\log(z):=\ln|z|+i\operatorname{Arg}(z)\tag{2}
$$
where $\arg(z)$ is the standard branch of the ...
- 928
3
votes
1
answer
131
views
How to Prove Convergence of $\prod\limits_{n =1}^\infty \left(1-\frac{z^2}{n^2}\right)$ for any $z \in \mathbb{C}$?
For $\frac{\sin(\pi z)}{\pi z} =\prod\limits_{n =1}^\infty \left(1-\frac{z^2}{n^2}\right)$, prove convergence of $\prod\limits_{n =1}^\infty \left(1-\frac{z^2}{n^2}\right)$ for any $z \in \mathbb{C}$.
...
- 6,620
0
votes
0
answers
31
views
Absolutely vs Non-Absolutely Convergent Infinite Product
Consider the following standard infinite product:
$$
\prod_{n=1}^{+\infty} \left( 1 + \frac{(-1)^n}{2n-1}z \right)
$$
This product is not absolutely convergent because:
$$
\sum_{n=1}^{+\infty} \left|...
- 1,179
0
votes
2
answers
58
views
Using Ratio Test for a Series with odd and even indexed coefficients
The problem I have is the following:
Suppose the radius of convergence of the series $\sum_{n=0}^\infty a_n z^n$ is equal to $2$. Find the radius of convergence of the series
$$\sum_{n=0}^\infty 2^{\...
- 415
10
votes
2
answers
377
views
Associativity of infinite products
It is well-known that if $\sum_{n=1}^\infty a_n$ is an absolutely convergent complex series and $\mathbb N$ is partitioned as $J_1,J_2,\dots$, then the series $\sum_{j\in J_n}a_j$ for all $n$ and $\...
- 1,434
2
votes
1
answer
33
views
Existence of uniformly convergent sequence of polynomials converging to an analytic function
Let $\Delta_1,\Delta_2,\dotsc,\Delta_n$ be mutually disjoint open discs in the complex plane.
Question part 1: Does there exists a sequence of polynomials $(p_n)_{n\in \mathbb N}$ converging uniformly ...
0
votes
0
answers
47
views
How to Show that Taylor Series of Holmorphic Function Centred at the Origin Converges Everywhere on the Open Unit Disk
I am interested in how one can show that the Taylor series of a holomorphic function defined within the open unit disk converges everywhere within the open unit disk.
To put this more clearly, suppose ...
4
votes
1
answer
117
views
Convergence of series from inverse of Cauchy product
The Cauchy product of two real or complex infinite series $\sum_{n\in\mathbb{N}} a_n$ and $\sum_{n\in\mathbb{N}} b_n$ is defined as:
$$ \forall n\in\mathbb{N}, c_n = \sum_{k=0}^n a_k b_{n-k} $$
...
- 3,815
0
votes
1
answer
25
views
How to construct a complex function series that converges uniformly on a closed region but its derivative does not converge uniformly on the boundary?
I'm looking for an example of a series of complex functions $\{f_n(z)\}$ with the following properties:
The series $\sum_{n=1}^{\infty} f_n(z)$ converges uniformly on a closed region $D$ in the ...
- 15
0
votes
0
answers
17
views
Convergence of Lesbegue measure of the images of a sequences converging uniformly
Good morning, my answer is related to the one introduced here
Convergence of images of a sequence of converging continuous functions
but with stronger hypotheses. Suppose you have a sequence of ...
0
votes
0
answers
37
views
Convergence of an aggregate remainder in a logarithmic expansion
Let $(x_j)_j$ be an ergodic and stationary sequence of random variables such that $E[x_j]=0$ (for all $j$) and:
$$\frac{1}{n}\sum_{j=0}^{n}f(x_j) \to E[f(x)] < \infty, \quad (n \to \infty)$$
for ...
- 75
1
vote
1
answer
99
views
How to show that $\sum_{\rho} 1/|\rho| = \infty$?
Just after Theorem 10.13 in the book Multiplicative number theory I: Classical theory by Hugh Montgomery, Robert C. Vaughan the following two statements are assumed to be without proofs. Perhaps they ...
- 281
2
votes
1
answer
67
views
Given a convergent sequence of complex numbers, is there a sequence of nested Jordan curves whose sum of members of $z_n$ in the regions converges?
Suppose $(z_n)_{n\in\mathbb{N}}\subset \mathbb{C}$ such that $\displaystyle\sum_{n=1}^{\infty} z_n$ converges.
Then, for example,
$$ \sum_{k=1}^{\infty}\left( \sum_{\frac{1}{k+1} < \left\lvert z_n \...
- 20.7k
1
vote
1
answer
123
views
Convergence of summation of complex exponentials with alternating exponent
Related to my previous question, consider $$f(s)=\sum_{k=1}^{\infty} \exp(-s(-2)^k)$$where $s\in\mathbb{C}$ is a complex number. According to the Willie Wong's comment, $f(s)$ diverges when $\Re\{s\} \...
- 4,359
1
vote
1
answer
74
views
Convergence of a limit as in Abels theorem
Abels theorem makes a statement about the continuity of a convergent power series for $x<1$
$$f(x)=\sum_{n=1}^\infty a_n x^n$$
at $x=1$, that is if $f(1)$ exists, it equals $\lim\limits_{x\...
- 6,277