All Questions
14
questions
3
votes
0
answers
64
views
Signed sum of sin and cos
In the study of graphical representations of the Ising model I have encountered the following sum for natural numbers $a,b$ such that $b \leq a$
$$
\sum_{\theta \in \{ \frac{ 2 \pi k}{q}, k = 0, \dots ...
3
votes
1
answer
262
views
On $\mathrm{\sum_{x\in\Bbb Z}sech(x)=3.142242…}$
Inspired by
This question,
I started to wonder about simpler series. I have seen similar questions to the following, but none had this special case explicitly. It is related to the q-digamma ...
- 12.4k
4
votes
0
answers
302
views
Closed form for Sum of Tangents with Angles in Arithmetic Progression
The formulae that can be used to evaluate series of sines and cosines of angles in arithmetic progressions are well known:
$$\sum_{k=0}^{n-1}\cos (a+k d) =\frac{\sin( \frac{nd}{2})}{\sin ( \frac{d}{2} ...
- 6,550
12
votes
1
answer
266
views
Could this conjecture be proved ? (sum of even powers of cotangents in arithmetic progression )
Having tried (in vain) to answer this question, I worked the explicit formulae of
$$\color{blue}{S_k=\sum _{n=1}^m \Big[\cot \left(\frac{n \,\pi }{2 m+1}\right)\Big]^k}$$ where $k$ is an even integer....
- 267k
0
votes
0
answers
66
views
Finding Closed Form Representation of A Sum of Trigonometric Functions
Is there a closed form representation of the sum $$\sum_{k=1}^x \cos\left(\frac{n\pi}{k}\right)$$ where $n$ and $x$ are integers? If not possible, is there a representation that uses special functions?...
- 788
2
votes
1
answer
163
views
Proving $\sum\limits_{k=1}^{N-1}\left[\frac{\sin\left(\frac{\pi km}{N}\right)}{\sin\left(\frac{\pi k}{N}\right)}\right]^{2}=m(N-m)$
I recently came across a sum (whose closed-form solution I was able to verify via Wolfram Alpha) but have no idea how to get there.
$$\sum\limits_{k=1}^{N-1}\left[\frac{\sin\left(\frac{\pi km}{N}\...
8
votes
1
answer
259
views
Showing that $\sum_{j=0}^{2n-1}{\cos^n(\frac{j\pi}{2n})(2\cos(\frac{2j\pi}n)+1)\cos(\frac{j\pi}2-\frac{2j\pi}n)}$ is never an integer for $n>10$
I want to show that
$$f(n) = \sum_{j=0}^{2n-1}{\cos^n\left( \frac{j \pi}{2n}\right) \left( 2\cos \left( \frac{2 j \pi}{n} \right) + 1\right) \cos \left( \frac{j \pi}{2} - \frac{2 j \pi}{n} \right)}$$
...
- 1,034
0
votes
0
answers
48
views
Coefficients of a polynomial with roots represented as squares of cosines
Consider a polynomial with roots $\cos^2 \left(\frac{j \pi}{2n + 1}\right), 1 \leq j \leq n$. The coefficients of this polynomial are sums of these numbers taken one at a time, two at a time, three at ...
4
votes
1
answer
263
views
How to calculate $\sum_{k=0}^n a^k\sin(kx)$?
I tried to evaluate
$$
\sum_{k=0}^n a^k\sin(kx)
$$
using complex numbers but it didn't work... Any hint?
$a$ and $x$ are real numbers.
- 145
4
votes
1
answer
417
views
Write $\sum_{k=1}^nk\sin(kx)^2$ in closed form
$\underline{Given:}$
Write in closed form $$\sum_{k=1}^nk\sin(kx)^2$$
using the fact that $$\sum_{k=1}^nku^k=\frac u{(1-u)^2}[(n)u^{n+1}(n+1)u^n+1]$$
$\underline{My\ Work:}$
I substituted $\sin(kx)^...
- 1,250
3
votes
3
answers
222
views
Closed form for a trigonometric partial sum
I know that:
$$\sum_{k=1}^n\arctan(2k^2)=\frac{\pi n}{2}-\frac{1}{2}\arctan(\frac{2n(n+1)}{2n+1})$$
Can a similar closed form expressions be given for $\sum_{k=1}^n \arctan(k^2)$?
I was able to ...
- 10.6k
8
votes
2
answers
685
views
Finite Series $\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})}$
I want to show that
$$\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})} = \frac{n^2-1}6$$
With induction I don't know how I could come back from $\frac{1}{1-\cos(\frac{2k\pi}{n+1})}$ to $\frac{1}{1-\...
- 81
51
votes
9
answers
6k
views
Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$
Question : Is the following true for any $m\in\mathbb N$?
$$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$
Motivation : I reached $(\star)$ by ...
- 144k
42
votes
3
answers
1k
views
Calculate the following infinite sum in a closed form $\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$?
Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.
$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
- 13.2k