0
votes
1answer
20 views

A question about the relation between divergence and absolute divergence.

Princeton Lectures in Complex Analysis by Stein and Shakarchi says the following: If $|z| > R$, then a similar argument proves that there exists a sequence of terms in the series whose ...
0
votes
2answers
46 views

Direct evaluation of a series from Euler's identity.

Is there a direct way to evaluate: $$ \sum_{k=0}^{\infty} (-1)^k \dfrac{\pi^{2k}}{(2k)!}=-1 $$ Note that this follows from Euler's identity.
1
vote
2answers
45 views

Sum of Complex series

Let $\theta\in\mathbb{R}$ and $\theta \neq k\pi$ for $k\in\Bbb Z$. By summing a geometric progression show that $$1 + e^{2i\theta} + e^{4i\theta}+e^{6i\theta} + e^{8i\theta}= ...
1
vote
2answers
61 views

When can an infinite sum and complex integral be interchanged?

Are there some conditions under which the following two are equal? $$\displaystyle \oint_C \sum f_n(z)= \sum \oint_C f_n(z)$$ In the case of real valued functions, the condition $f_n(z) \geq 0$ ...
0
votes
0answers
15 views

Convergence of Series of Complex Numbers with Decreasing Modulo (non-zero imaginary part)

Let $(a_n)_{n \in \mathbb{N}}$ be a decreasing sequence of positive real numbers tending to zero. Show that for $\theta \in \mathbb{R}$, $\theta$ not a multiple of $2\pi$, the series $\sum_{n\geq1} ...
2
votes
1answer
38 views

Complex Analysis Weierstrass M-Test

Prove that each of the following series converges uniformly on the corresponding subset of $\mathbb C$: $$\begin{align*} \text{(a)} \; & \sum_{n=1}^\infty \frac{1}{n^2 z^{2n}}, & & ...
2
votes
0answers
49 views

$f\in C^\omega ((a-R,a+R),\mathbb{R})$ [closed]

We discuss the following question in the field of real numbers . A a power series $f(x)= \sum_{n=0}^\infty a_n (x-a )^n$ converges in $(a-R,a+R).$ Prove: $$\forall x_0\in\left(a-R,a+R \right), ...
1
vote
1answer
30 views

Analytic Extension: Imaginary Stripe

I was always wondering the following: Given a real analytic function there exists a positive radius of convergence for every point. This won't be affected by allowing complex numbers so it extends ...
1
vote
1answer
22 views

Computing the radius of convergence of a given series

Could anyone help me with the following problem? I'm getting $1$ as the answer, but I found the solution (without justification) to this problem online and it says the answer is $1/2$. Determine ...
1
vote
1answer
30 views

Radius of convergence of series given radius of convergence of another series

I'm hoping someone might be able to verify my solution to the following problem: Suppose that the series $\sum c_n z^n$ has radius of convergence $R$. Find the radius of convergence of the $\sum ...
1
vote
1answer
68 views

Help on a tough summation from Rudin?

I'm having a tough time deriving (4) from the bracketed expression in (3) shown in the photo. I've been futzing with partial sums of geometric series and binomial expansions for a while now with no ...
4
votes
1answer
95 views

What is $\zeta(n)$ as $n$ tends to $\infty$? How fast it goes to the limit?

What is $\zeta(n)$ as $n\to\infty$? How fast it goes to the limit?
2
votes
1answer
76 views

Power series difficulty

How would I find the region of convergence of the series of $\frac{1}{n^3}(\frac{z+1}{z-1})^n$. I thought about rewriting $\frac{z+1}{z-1}$ as $\frac{2}{z-1}+1$ but I don't think that helps. Thanks
0
votes
2answers
28 views

Radius of convergence query

Find the radius of convergence of the series of $\frac{2^n(4z-8)^n}{n}$ My answer: $(4z-8)^n=4^n(z-2)^n=2^{2n}(z-2)^n$. Let $c_{n}=\frac{2^{3n}}{n}$. Then $\frac{c_{n}}{c_{n+1}}=\frac{n+1}{2n}$ so ...
6
votes
2answers
154 views

From the series $\sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)$ to $\zeta(\frac{1}{2}+it)$

Here is a pretty series $$ \displaystyle \sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)=\frac{1}{2} \left(1-\ln (2\pi)+\gamma\right) \quad (*) $$ where $H_{n}:=\sum_{1}^{n} ...
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 ...
1
vote
0answers
12 views

Taylor's expansion of the singular part of an analytic function

Assume $f$ is analytic on the annulus $R_1<|z-a|<R_2$. Assume $R_1<r<|z-a|$. Define $f_2$ by $$f_2(z)=\frac1{2\pi i}\int_{|x-a|=r}\frac{f(x)dx}{x-z}$$ $f_2$ is analytic on $|z-a|>r$. ...
0
votes
2answers
38 views

solving $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}|$ graphicly [closed]

how to solve $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}| $ in the graphic method?
2
votes
0answers
68 views

What is the limit of this sequence of complex numbers?

Let $z_1$ and $z_2$ be two complex numbers in the upper half-plane. Does the sequence $c_n = \exp^n\left(\sqrt{\log^n(z_1)*\log^n(z_2)}\right)$ converge to a fixed point as $n\to\infty$? If so, what ...
4
votes
5answers
239 views

How to find $\sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)}$

Let $a,b$ be two unequal integers. I have to find the sum below. $$ \sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)} $$ I should use complex analysis, but I have no clue where to start. I only now that I can ...
1
vote
0answers
27 views

Exchanging Limits in Series inversion

I have the Lagrange Bürmann formula as follows: $$\sum^{m-1}_{n=1}\frac{1}{n!}(w-b)^n\lim_{z \rightarrow a} \frac{d^{n-1}}{dz^{n-1}} \left(f^{'}(z)\frac{z-a}{g(z)-g(a)}\right)^n$$ Where g(z) is the ...
3
votes
1answer
79 views

Fourier series without Fourier analysis techniques

It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula ...
1
vote
2answers
30 views

Limit of complex numbers' sequence (related to Möbius transformation)

Problem Let $T(z)=\dfrac{7z+15}{-2z-4}$. Let $z_1=1$ and $z_n=T(z_{n-1})$ for $n\geq 2$ Find $\lim_{z_n \to \infty}z_n$ I am having a lot of difficulties trying to solve this. I've tried to find a ...
1
vote
0answers
59 views

how to prove $\sum_n |b_n|^2<\infty$

$\{b_n\}$ is a complex sequence, If for all $\ell^2$ sequences $a_n$, we have $\sum_n \bar{a}_nb_n$ converges . Prove that $\sum_n|b_n|^2<\infty$
1
vote
2answers
39 views

Equivalent metrics in $\mathbb C$ (via stereographic projection)

Problem statement Let $\overline{d}$ be the distance in $\bar{\mathbb C}$ induced by the distance of $\mathbb R^3$ through the stereographic projection, i.e., if $z,z' \in \bar{\mathbb C}$, we define ...
0
votes
0answers
34 views

series functions of complex variable $ z $ and alternating zeta function convergence

Let $f_n(z)$ and $p_n(z)$ two series functions of complex variable $z$ defined as the following: $f_n(z)$=$ \sum_{n}exp({(-1)}^{n-1}{n^{-z}})) $ $p_n(z)$=$exp(\sum_{n}({(-1)}^{n-1}{n^{-z}})$ ...
1
vote
0answers
20 views

How to characteristize or distinguish those lacunary series with same natural boundary from each other

There are lots of different lacunary series with integral coefficients and with same natural boundary. As we know,Some functions can be distinguished or characterized by their poles. Now,the ...
1
vote
1answer
36 views

Laurent series of $\frac{e^{iz}}{z^2+p^2}$, $ p>0$.

I need help finding the main part of the laurent series of $f(z)=\frac{e^{iz}}{z^2+p^2}$ in $ip,-ip$ since these are the two poles of $f$. Due to the orders of the poles are 1 I just have to find ...
1
vote
1answer
33 views

Show that $\sum_{k=1}^N\frac{1}{(k+a)(k+b)}=\frac{1}{b-a}\sum_{a<k\leq b}\frac{1}{k}-\frac{1}{b-a}\sum_{a<k\leq b}\frac{1}{k+N}$

I am quite stuck on this problem and I don't know how to proceed. The question states: Let $a,b,N\in\mathbb{N}$, $b>a$, $N\geq b-a$. Show that ...
2
votes
1answer
38 views

If $\sum_{n=1}^\infty |a_n|^2<\infty$, Then : $\sum_{n=1}^\infty a_n$ Converges $\Leftrightarrow \prod_{n=1}^\infty(1+a_n)$Converges

Prove that for complex sequence $\{a_n\}_{n\in\mathbb{N}}$ : if $\displaystyle \sum_{n=1}^\infty |a_n|^2<\infty$, Then : $\displaystyle \sum_{n=1}^\infty a_n$ Converges $\Leftrightarrow ...
1
vote
1answer
35 views

Divergent Complex sequence with distinct points and without accumulation point tends to $\infty$?

Is it true that any Divergent complex sequence with distinct points and without accumulation point tends to $\infty$ ? (One can also replace distinctness condition by condition finitely repeating ...
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
0answers
39 views

Change of Double Sum to Single Sum for Large N

I am reading article on X-ray interference. In the article authors claim that it is possible to replace double sum $\frac{1}{N} \sum_{n=1}^{N} \sum_{k=1}^{N} e^{-i \phi_{n}} e^{i \phi_{k}}$ by ...
5
votes
4answers
160 views

How to find the following sum? $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{4n + 9}} - \frac{1}{{4n + 7}}} \right)} $

I want to calculate the sum with complex analysis (residue) $$ 1 - \frac{1}{7} + \frac{1}{9} - \frac{1}{{15}} + \frac{1}{{17}} - ... $$ $$ 1 + \sum\limits_{n = 0}^\infty {\left( {\frac{1}{{4n + 9}} - ...
1
vote
1answer
32 views

Is it correct to say that this series diverge in the given annulus?

Let's consider the annulus $1\le|z|\le2$ and $$f(z)=\frac{2z+3}{z^2+3z+2}$$ Find its Laurent series. First I simplify the expresion: ...
0
votes
1answer
25 views

Find Laurent's series of these two functions around $z_o$

Find the Laurent series of $f(z)=\frac{z}{(z+1)^2}$ around $z_o=-1$, and $g(z)=z\exp(\frac1{z+i})$ around $z_o=-i$. For $f$, what they're asking is to find the series in $0<|z+1|$. On the ...
1
vote
2answers
32 views

a series derived from a holomorphic function converges implies that the coefficients converge to $0$

Let $D=\{z\in\mathbb{C}\mid |z|<2\}$. Let $f:D\setminus\{\frac{i}{2}\}\longrightarrow \mathbb{C}$ be holomorphic with $f(z)=\sum_{n=0}^\infty a_nz^n$ for any $|z|<\frac{1}{2}$. Suppose $a_n\neq ...
2
votes
1answer
66 views

series of an arbitrary sequence multiplying 1/n

Let $(r_n)_{n=1}^\infty$ be an arbitrary sequence of numbers in $[0,1]$. The series $\sum_{n=1}^\infty\frac{1}{n^2\sqrt{|x-r_n|}}$ converges for almost all $x$ in $[0,1]$. Is it true or not true? I ...
1
vote
4answers
773 views

Lagrange's Trigonometric Identity

Lagrange's Trig identity is $$ 1+\cos\theta+\cos 2\theta +\cdots + \cos n \theta=\frac{1}{2}+\frac{\sin\frac{(2n+1)\theta}{2}}{2\sin \frac{\theta}{2}},\quad (0<\theta <2\pi). $$ How can we prove ...
1
vote
0answers
25 views

How do we show $\ln z=\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}(z-1)^n$ for all $z\in\mathbb{C}$ with $|z-1|<1$?

Let $$g:B_1(1):=\left\{z\in\mathbb{C} :|z-1|<1\right\}\to\mathbb{C}\;,\;\;\;z\mapsto\ln z-\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}(z-1)^n$$ (1) In a first step, I'm asked to show, that $g$ is ...
1
vote
1answer
25 views

sequence of analytic functions on an open subset of $\mathbb{C}$ that converges uniformly on compact subsets

Let $U$ be an open subset of $\mathbb{C}$ and let $\{f_n\}$ be a sequence of analytic functions on $U$. Suppose that $(f_n)$ converges uniformly on any compact subsets of $U$ to a function $f$. Let ...
1
vote
0answers
78 views

Complex Analysis: Laurent Series Expansion

I am given the following. Let $f(z) = \frac{e^z-e^{-z}}{z^4}$. Give a Laurent Series for $f(z)$ in terms of powers of $z$. For which values of $z$ does the Laurent series converge? Justify ...
0
votes
1answer
35 views

Finding Region of Convergence

I am trying to find the regions of absolute and uniform convergence for three different series, but I figured I'd start with the simplest one $$\sum_{k\geq 2} k(k-1)z^{k-2}$$ I have worked with ...
1
vote
1answer
52 views

Help with proving that this series diverges

I have this complex series $\sum\limits_{n=1}^{\infty} \frac{i^n}{\sqrt{n}}$, which I'm trying to prove converges. Now, I know that a complex sequence converges iff both its real and its imaginary ...
1
vote
1answer
47 views

using Maclaurin series to estimate $\frac{1}{e} $

The problem I'm trying to solve is: Determine how many terms of the Maclaurin series of $f(x) =e^{-x}$ should be used to estimate $\frac{1}{e} $ with an error of magnitude less than $5 \times ...
0
votes
1answer
72 views

Convergence of $\frac{1}{2^k} \frac{1}{z-w_k}$

Suppose $w_1,w_2,w_3,...$ are points on the unit circle. Consider the infinite series $$\sum_{k=1}^{\infty} \frac{1}{2^k} \frac{1}{z-w_k}$$ Let $D=\{z \in \mathbb{C}: |z|<1 \}$ A) Show that series ...
0
votes
0answers
128 views

A uniformly convergent series

How does one show that the series $$\sum_{k = 1}^\infty \left\{\frac{s}{k} - \log\left(1 + \frac{s}{k}\right)\right\}, \quad s \in \mathbb{C} \setminus \{0, -1, -2, \ldots\}$$ is uniformly convergent? ...
1
vote
1answer
56 views

$\frac{1}{(1+s^{2}) (1+t^{2})}$ real analytic in $\mathbb R^{2}$ but not real-entire; why?

A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point $(s_{0}, t_{0})$ in there corresponds an expansion with ...
1
vote
0answers
52 views

Closed form of $\sum_{n=1}^\infty(-1)^{n+1} \frac xn \ln\left(1+\frac xn\right), \quad x \in (0,1)$

Is there a known closed form of the series below? $$\sum_{n=1}^\infty(-1)^{n+1} \frac xn \ln\left(1+\frac xn\right), \quad x \in (0,1]$$
1
vote
0answers
29 views

Two series with the same radius of convergence (Lang's exercise)

In Lang's Complex Analysis, there is an exercise that says: Let $f(z)=\sum a_nz^n$ have radius of convergence $r>0$. Show that the series $\sum n^da_nz^n$ has the same radius of convergence. How ...