All Questions
28
questions
3
votes
1
answer
82
views
Does $\sum_{j=1,j\neq k}^n\frac{z_k}{z_k-z_j}=\sum_{j=1,j\neq m}^n\frac{z_m}{z_m-z_j}$ implies the $(z_j)_{j=1,...,n}$ are the nth roots of $z_1^n$?
Let $(z_1,\ldots z_n)\subseteq \mathbb{C}.$
Does
$$
\sum_{j=1\,,\,j\neq k}^n \frac{z_k}{z_k-z_j}=\sum_{j=1\,,\,j\neq k'}^n \frac{z_{k'}}{z_{k'}-z_j}
$$ for all $k,k'\in\{1,...,n\}\,$, implies that ...
- 87
0
votes
1
answer
46
views
Find $\sum_{j = 1}^{2004} i^{2004 - F_j}$ where $F_n$ is the nth Fibonacci number
The Fibonacci sequence is defined by $F_1 = F_2 = 1$ and $F_n = F_{n - 1} + F_{n - 2}$ for $n \ge 3.$
Compute
$$\displaystyle\sum_{j = 1}^{2004} i^{2004 - F_j}.$$
I tried computing the first few ...
- 800
0
votes
0
answers
148
views
How can I solve this two-sided infinite summation?
In this question, the comment suggests that the imaginary terms in the stated solution might sum to $0$. In order for this to happen, it must be the case that
$\sum_{-\infty}^\infty [\frac{2(-1)^{n + ...
- 1,912
1
vote
1
answer
194
views
Trigonometric Identities Using De Moivre's Theorem
I am familiar with solving trigonometric identities using De Moivre's Theorem, where only $\sin(x)$ and $\cos(x)$ terms are involved. But could not use it to solve identities involving other ratios. ...
0
votes
2
answers
177
views
Sum of the roots of unity, $z_{1}^p+...+z_{n}^p$ [duplicate]
Let $z_1,...,z_n$ be the $n$ roots of unity. I am not able to find a value for the sum: $$z_{1}^p+...+z_{n}^p,\ p \in \Bbb N$$
I know that this sum can also be written as $$\sum_{k=0}^{n-1}e^{i(\frac{...
- 35
10
votes
2
answers
417
views
If $z\in\mathbb C$ with $|z|\leqslant\frac{4}{5}$, then $\sum_{n\in S}z^n\neq-\frac{20}{9}$
Let $z$ be a complex number with $|z|\le\tfrac{4}{5}$. If $S\subset\mathbb N^+$ is a finite set, then I'd like to show that
$$\sum_{n\in S}z^n\neq-\frac{20}{9}\,.$$
This problem is from an exam in ...
- 93.6k
2
votes
1
answer
86
views
Find the sum of infinite series $\cos{\frac{\pi}{3}}+\frac{\cos{\frac{2\pi}{3}}}{2}+..$
Find the sum of infinite series $$\cos{\dfrac{\pi}{3}}+\dfrac{\cos{\dfrac{2\pi}{3}}}{2}+\dfrac{\cos{\dfrac{3\pi}{3}}}{3}+\dfrac{\cos{\dfrac{4\pi}{3}}}{4}+\dots$$
I tried to convert it to $\mathrm{cis}$...
- 1,157
3
votes
4
answers
174
views
Evaluate $\frac{q}{1+q^2}+\frac{q^2}{1+q^4}+\frac{q^3}{1+q^6}$, where $q^7=1$ and $q\neq 1$.
Let $q$ be a complex number such that $q^7=1$ and $q\neq 1$. Evaluate $$\frac{q}{1+q^2}+\frac{q^2}{1+q^4}+\frac{q^3}{1+q^6}.$$
The given answer is $\frac{3}{2}$ or $-2$. But my answer is $\pm 2$.
...
- 373
1
vote
1
answer
161
views
Summation of rational complex function
Let $z$ be a non-real complex number such that $z^{23}=1$. Compute $$\sum_{k=0}^{22} \frac {1}{1+z^k +z^{2k}}$$.
I could not really do much about this problem. I tried writing the summand as a ...
- 407
3
votes
2
answers
95
views
Complex numbers algebra problem involving cyclic summation
Let $a_1$, $a_2$, $a_3\in \mathbb{C}$ and $|a_1|=|a_2|=|a_3|=1$.
If $\sum\frac{a_1^{2}}{a_2 a_3}=-1$, find $|a_1 + a_2 + a_3|$
What I have done till now:
First, I tried to attack the required sum ...
- 407
-1
votes
1
answer
27
views
Query about summation in derivation of complex fourier series
I was trying to follow a derivation on the complex fourier series, but I am a bit confused at one particular step. In the following video https://www.youtube.com/watch?v=Ft5iyapkSqM, at 6:30, the ...
- 275
0
votes
2
answers
78
views
Evaluating $\frac{\sum_{k=1}^{1010} i^{2k-1}}{\prod_{k=1}^{1010} i^{2k}}$, where $i$ is the imaginary unit [closed]
Can somebody help me evaluate the following?
$$\frac{\sum_{k=1}^{1010} i^{2k-1}}{\prod_{k=1}^{1010} i^{2k}}$$
Where $i$ is the imaginary unit: $i^2 = -1$.
Edit:
Thanks for helping with formatting ...
6
votes
4
answers
3k
views
if 1, $\alpha_1$, $\alpha_2$, $\alpha_3$, $\ldots$, $\alpha_{n-1}$ are nth roots of unity then...
if 1, $\alpha_1$, $\alpha_2$, $\alpha_3$, $\ldots$, $\alpha_{n-1}$ are nth roots of unity then
$$\frac{1}{1-\alpha_1} + \frac{1}{1-\alpha_2} + \frac{1}{1-\alpha_3}+\ldots+\frac{1}{1-\alpha_n} = ?$$
...
- 1,396
1
vote
3
answers
122
views
Evaluating $\ i+i^2+i^3+i^4+\cdots+i^{100}$
$$i+i^2+i^3+i^4+\cdots+i^{100}$$
I figured out that every four terms add up to zero where $i^2=-1$, $i^3=-i$, $i^4=1$, so
$$i+i^2+i^3+i^4 = i-1-i+1 = 0$$
Thus, the whole series eventually adds up to ...
0
votes
1
answer
64
views
Sum containing $i$ and Fibonacci sequence
If the Fibonacci sequence is defined such that $F_1=F_2=1$, compute $$\sum_{j=1}^{2012}i^{2012-F_j}$$ where $i$ is the imaginary unit.
I tried writing out the terms and using the laws of exponents ...
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