1
vote
2answers
38 views

Sequence of alternating $0$'s and $1$'s in terms of $i$?

How to redefine the function $f(n) = \begin{cases} 1, & \text{if $n$ is even} \\ 0, & \text{if $n$ is odd} \end{cases}$ in terms of arithmetic operations using ⅈ?
0
votes
0answers
12 views

If $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$

I tried proving that if $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$. This is my attemt. Let $\epsilon>0$. Then there exists $N\in ...
1
vote
3answers
79 views

Proof of $\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos n\theta=\frac{\sin\frac12n\theta}{\sin\frac12\theta}\cos\frac12(n+1)\theta$

State the sum of the series $z+z^2+z^3+\cdots+z^n$, for $z\neq1$. By letting $z=\cos\theta+i\sin\theta$, show that $$\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos ...
0
votes
0answers
21 views

hyperbolic series $\sum_{r=1}^n \cosh(rx)$

I have attempted to do this question on hyperbolic functions: Prove that $$\cosh x + \cosh 2x + ... + \cosh nx ...
0
votes
0answers
42 views

complex limits, how to show they go to 0?

In complex integration my book uses that some limits go to zero as R goes to infinity. However I do not now how to show this, these two limits are: $e^{-\pi(R^2+2iRy-y^2)}$, where y is a real number ...
0
votes
3answers
54 views

Find the sum $1+\cos (x)+\cos (2x)+\cos (3x)+…+\cos (n-1)x$ [duplicate]

By considering the geometric series $1+z+z^{2}+...+z^{n-1}$ where $z=\cos(\theta)+i\sin(\theta)$, show that $1+\cos(\theta)+\cos(2\theta)+\cos(3\theta)+...+\cos(n-1)\theta$ = ...
5
votes
0answers
139 views

Convergence of infinite series of complex numbers [duplicate]

This has been bugging me for some months since our lecturer, a fields medalist, mentioned that he couldn't solve it when he was our age, yet had had two students submit solutions to it (during our ...
1
vote
2answers
72 views

If a series converges then the power series converges for all z

How can I prove that if $\sum \limits_{n=1}^{\infty} c_n$ , $c_n\in \mathbb{C}$, converges then $\sum \limits_{n=1}^{\infty} c_n \frac{z^n}{1-z^n}$ converges for all z in $\mathbb{C}$ with ...
3
votes
1answer
56 views

Find a sequence

Find the function for the sequence $a_0 = 0, a_1 = 1$ and $a_{n}=a_{n+10}+a_n$ for all $n>0$.
2
votes
2answers
107 views

Complex series radius convergence

How to find the values for which $z$ converges, $z\in\mathbb{C}$, in the serie $$\sum_{n=1}^{\infty}\frac{1}{(1+|z|^{2})^{n}}$$ I know I have to use the convergence radius expression, but what I ...
0
votes
3answers
61 views

On a certain series of complex numbers

Is it possible that the above infinite series is equal to ?
3
votes
1answer
82 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
35 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 ...
0
votes
1answer
46 views

Proving the Mandelbrot set is bounded

I am trying to prove that the Mandelbrot set defined as the set $\mathcal M$ of complex numbers $c$: the recursive sequence defined as $$z_0=c, \space \space \space z_{n+1}={z_n}^2+c$$ is bounded. ...
1
vote
4answers
66 views

$ \sum\limits_{n=1}^\infty \dfrac{(1+i)^n}{n^2}$ is divergent, and no idea about $\sum\limits_{n=1}^\infty \dfrac{(3+4i)^n}{5^n\,\sqrt[999]{n}}$

How can one see that $ \sum\limits_{n=1}^\infty \dfrac{(1+i)^n}{n^2}$ is divergent, and by which criterion? I was using a binomial theorem for $ (1+i)^n $ as $ \sum\limits_{k=0}^n \dbinom{n}{k} i^n$, ...
0
votes
1answer
61 views

A complex series with exponentials

I have tried to solve this type of series : $$\sum \frac{e^{i\, u(n)}}{v(n)} $$ For some $u,v$ an Abel Transform allow to find convergence, but for $u(n)=n^2$ and $v(n)=n$ I can't find an argument. ...
1
vote
0answers
16 views

When is $\sum_{n,m=-\infty}^\infty \frac{1}{(n\omega_1+m \omega_2)^\alpha}\in \mathbb{R}$?

This came up when reading about elliptic functions, where $\frac{\omega_1}{\omega_2}\notin\mathbb{R}$, and $\alpha>2$ for $$S(\omega_1,\omega_2,\alpha)=\sum_{\begin{matrix} n,m=-\infty\\ (n,m)\ne ...
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? ...
0
votes
1answer
37 views

$\sin(1+\frac{1}{z-1})$ expanded in powers of $z-1$

The whole problem: Obtain the following Laurent expansions. State the first four nonzero terms. State explicitly the $n$th term in the series, and state the largest possible annular domain in which ...
2
votes
1answer
74 views

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 ...
0
votes
2answers
65 views

Uniform convergence of the series

Test the uniform convergence of the series $$ \sum_{n=1}^\infty \frac{1}{z^2 - n^2 \pi^2}$$ $$ \forall z \not= \pm n\pi,\;\; where n \in\mathbb N$$ Can I find $M_n$ such that $$ ...
1
vote
0answers
43 views

Test the uniform convergence of the series in indicated region

Test the uniform convergence of the series I tried to find $M_n$ such that $|\sum_{n=1}^ \infty(-1)^n\frac{z^{2n-1}}{1-z^{2n-1}}|\le M_n $ by using Abel's Theorem This is the question : Test the ...
0
votes
0answers
25 views

Summing complex numbers of magnitude $1$

A professor at my university gave me the following problem: For what real values of $m$ do we have $$L=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^Ne^{2\pi ik^3m}=0$$ If $m$ is rational, expressed ...
1
vote
1answer
46 views

Proof complex series

I have to prove this: $\displaystyle\sum_{n=1}^\infty n\alpha^n = \displaystyle\frac{\alpha}{(1-\alpha)^2}$ if $|\alpha | < 1$ I think this is a geometric series, and i have to solve it with a ...
2
votes
2answers
115 views

Sum of $\sum\limits_{n=1}^\infty q^n \sin(nx)$

How to find $\sum\limits_{n=1}^\infty q^n \sin(nx)$, where $|q|<1$ and $x \in \mathbb{R}$? I was thinking about rewriting it as $\sum\limits_{n=1}^\infty (q(\Im(\cos x+i\sin x)))^n$. It is a ...
3
votes
2answers
362 views

Proof of $\sin^2(x) + \cos^2(x)=1$ using series

I have to prove the following identity $\sin^2 (x) + \cos^2(x)=1$. I can easily prove this, but this exercise is given in the section introducing the series expansions for $\sin(x)$ and $\cos(x)$ and ...
3
votes
1answer
32 views

Effect of sigmas inequality on sequences

We have two nets of complex numbers $\{z_\alpha\}_{\alpha\in I},\{w_\alpha\}_{\alpha\in I}$ for some set $I$ which might be uncountable, and we have $$\sum_{\alpha\in I}|w_\alpha|\leq ...
0
votes
0answers
32 views

Is there a way to expand Re(Li(a^z)) in series?

I'm searching a way to expand $ f(z) = Re(Li(a^z)), a \in R, z \in C $ in series. The computer-friendly, quickly convergent series is a huge plus. For being 'computer-friendly' I mean a relatively ...
0
votes
1answer
15 views

Expressing array response $A(Z) = \sum_{-N}^{N} w_n Z^n$ as sine-function

The array-response of an antenna can be defined as: $$A(Z) = \sum_{-N}^{N} w_n Z^n$$ where $Z = \exp(-i \omega \Delta t) = \exp(-ik\Delta x \sin \alpha)$ According to my textbook, if we let $w_n = ...
6
votes
4answers
94 views

$\frac{1}{1-e^{\frac{ik\pi}{n+1}}}+\frac{1}{1-e^{-\frac{ik\pi}{n+1}}}=1$?

I'm working on an assignment where part of it is showing that $S_k=0$ for even $k$ and $S_k=1$ for odd $k$, where $$S_k:=\sum_{j=0}^{n}\cos(k\pi x_j)= \frac{1}{2}\sum_{j=0}^{n}(e^{ik\pi ...
3
votes
3answers
133 views

Radius of convergence of the series $\displaystyle\sum\limits_{n=0}^\infty \frac{n!\,z^{2n}}{(1+n^2)^n}$

I am doing the following problem and would like to know whether my answer is correct or not: Find the Radius of convergence for the complex series $\displaystyle\sum\limits_{n=1}^n ...
1
vote
2answers
314 views

Laurent series for $(\sin 2z)/z^3$

I have to find the Laurent series for $(\sin2z)/z^3$ in $|z|>0$, but I really don't know how to start. And I thought that in this area it's a Taylor serie because the singularity isn't in the area, ...
1
vote
2answers
102 views

Laurent Series Difficulty

Hello all at StackExchange, I'm having some trouble understanding computing the Laurent series for different domains. Here's my approach to finding the Laurent series for $\dfrac{3}{(z+1)(z-2)}$ for ...
3
votes
2answers
36 views

Convergence of $\sum_{n=2}^{\infty}n^2\left(\frac{1-i}{2+i}\right)^n$

Does this sequence converge/diverge and if so, does it in a (not)absolute way? $$\sum_{n=2}^{\infty}n^2\left(\frac{1-i}{2+i}\right)^n$$
3
votes
1answer
58 views

Convergence of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ If $z=re^{i\theta}=x+iy$, $$F(z)=\sum_{n=1}^\infty r^na_ne^{in\theta}$$ is a harmonic ...
1
vote
0answers
61 views

Can't match boundary conditions on a perturbation series solution to a non-linear ODE?

I'm trying to generate a naive perturbation series solution (with all associated secular terms included) to the Rayleigh equation: \begin{equation} \frac{d^2y}{dt^2} + y = \epsilon ...
1
vote
1answer
67 views

FTOA: Let $f(z) = |a_nz^n + .. + a_1z + a_0|$. Show $f(z)$ has a minimizer $z^*$ and complex sum converges for $|z| \rightarrow \infty$

I've been looking into this proof of the Fundamental Theorem of Algebra: http://cuhkmath.wordpress.com/2011/06/28/another-proof-of-the-fundamental-theorem-of-algebra/ In the proof we have $f(z) = ...
4
votes
1answer
47 views

Calculation of $\sum_{k=0}^\infty \sin(x)^{ki}$

Is this formula correct? $$\sum_{k=0}^\infty \sin(x)^{ki}=\frac{i}{\sin(\ln(\sin(x)))-i\cos(\ln(\sin(x)))+i}$$ How is it possible to give a proof of this equality? Thanks.
0
votes
1answer
45 views

Why $\lim_{n\to \infty} ({3^n+2^n\over 3^n-2^ni^n}) = \lim_{n\to\infty} ({3^n+2^n\over 3^n-2^n}) \ $?

This week I was introduced to the limits of complex sequences. It is actually pretty simple because it's mostly the same compared to real sequences. However, there is one thing - Why is: ...
1
vote
1answer
34 views

Mathematical series regarding complex (I think)

$\sum _{k=1}^{n-1} (n-k)\cos\frac{2k\pi}{n} $ I smell complex here...something regarding $n^{th}$ roots of unity... But I think there might be a catch...after all: ...
0
votes
3answers
558 views

How to find the sum of a geometric progression involving cos using complex numbers?

Use $ 2\cos{n\theta} = z^n + z^{-n} $ to express $\cos\theta + \cos3\theta + \cos5\theta + ... + \cos(2n-1)\theta $ as a geometric progression in terms of $z$. Hence find the sum of this progression ...
1
vote
2answers
120 views

If a sequence of summable sequences converges to a sequence, then that sequence is summable.

Let $(a_i)^n$ be a sequence of complex sequences each of which are summable (they converge). Then if they have a limit, the limit sequence $(b_i)$ is also summable. All under the sup norm for ...
0
votes
1answer
39 views

complex sequences

my series and sequence knowledge has gone a little rusty so I was wondering if you could help me on the right path here. The assignment is to calculate the sum of the series (1/8)^n * e^(j(npi)/8) as ...
8
votes
3answers
182 views

Proving that the limit of a sequence is $> 0$

Let $u$ be the complex sequence defined as follows : $u_0=i$ and $ \forall n \in \mathbb N, u_{n+1}=u_n + \frac {n+1-u_n}{|n+1-u_n|} $ . Consider $w_n$ defined by $\forall n \in \mathbb ...
4
votes
2answers
208 views

How to evaluate a zero of the Riemann zeta function?

Here is a super naive question from a physicist: Given the zeros of the Riemann zeta function, $\zeta(s) = \sum_{n=1}^\infty n^{-s}$, how do I actually evaluate them? On this web page I ...
7
votes
3answers
195 views

Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$

I'm going to teach a preparation course for the complex analysis qualifying exam from my university (which basically consists of me doing some problems from past exams) and I'm trying to solve some ...
4
votes
1answer
108 views

Prove a Trigonometric Series

Question: $$\cot^{2}\frac{\pi }{2m+1}+\cot^{2}\frac{2\pi }{2m+1}+\cdots+\cot^{2}\frac{m\pi }{2m+1}=\frac{m(2m-1)}{3}$$ $m$ is a positive integer. Attempt: I started by showing that ...
7
votes
1answer
93 views

Why does $\sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )+{1\over2}-{1\over3} = \gamma$?

How could one prove that $$x = \sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )$$ is such that $x+{1\over2}-{1\over3} = \gamma$ ? I am having problems just calculating ...
1
vote
4answers
147 views

Evaluation of a complex numbers partial sum

Let $w = e^{i\frac{2\pi}5}$. I would like to evaluate $$w^0 + w^1 + w^2 + w^3 +...+ w^{49}$$ Can anyone please give me an idea how to evaluate the expression? Thanks in advance
8
votes
5answers
264 views

When are we (not) allowed to replace $x$ by $ix$?

It seems to be quite a common manipulation to replace $x$ by $ix$. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...