Tagged Questions
12
votes
3answers
150 views
$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$
Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.
$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
3
votes
1answer
62 views
Proof the following trig series
Prove that
$$\frac{ \sin x}{ \cos x}+\frac{\sin2x}{\cos^{2}x}+\frac{\sin3x}{\cos^{3}x}+\cdots+\frac{\sin nx}{\cos^{n}x}=\cot x-\frac{\cos(n+1)x}{\sin x \cos^{n}x}$$
I am not necessarily looking for a ...
7
votes
1answer
89 views
Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$
I need help with calculating this sum:
$$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
3
votes
1answer
44 views
Show that the following product equals 1 (involves trig)
How can I show that:
$$\prod_{k=1}^{n}\left ( 1+2\cos\frac{2\pi .3^{k}}{3^{n}+1} \right )=1$$
Could you please explain to me how to approach this problem?
Thank you.
12
votes
2answers
119 views
Proving the inequality $\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin(1/k^2)}{\cos^2 (1/(k+1))}$
How am I supposed to prove this inequality?
$$\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin\left(\frac{1}{k^2}\right)}{\cos^2 \left(\frac{1}{k+1}\right)}$$
Jordan inequality might be an option but led me ...
2
votes
2answers
37 views
Prove that a sum converges to a trigonometric expression
$$2^n \cos \left (\frac{n \pi}{2} \right )=\sum_{k=0}^{n} (-1)^k \binom{2n}{2k}$$
I expanded the LHS and got $$\binom{2n}{0}-\binom{2n}{2}+\binom{2n}{4}-\binom{2n}{6}+\cdots+(-1)^{n}\binom{2n}{2n}$$
...
13
votes
3answers
218 views
Proving a trig infinite sum using integration
How can I prove the following using integration and elementary functions?
Prove that:
$$\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{n} = \frac{\pi}{2} - \frac{\theta}{2}$$
$0 < \theta < 2\pi$
7
votes
5answers
240 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 ...
0
votes
1answer
77 views
Prove this proprety of $f(x)$
I've asked this question before a long time ago, but I didn't get a complete answer. This is the link to the incomplete answer: Prove the following property of $f(x)$?
Let ...
2
votes
2answers
64 views
$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$
$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$.
1
vote
2answers
59 views
Finding an infinite trigonometric sum
Find the following infinite sum : $$q\sin a+q^2\sin 2a+\ldots+q^n\sin na+\ldots$$ where $|q|<1$ .It would be good if you could find it without the help of any auxiliary sequences using only ...
4
votes
3answers
94 views
How do we find specific values of sin and cos given the series definition
$\exp:x\mapsto \sum\limits_{n=0}^{+\infty}\cfrac{1}{n!}x^n$
$\cos:x\mapsto \Re\left(\exp \left(i x\right)\right)=\sum\limits_{n=0}^{+\infty}\cfrac{\left(-1\right)^n}{\left(2n\right)!}x^{2n}$
...
1
vote
0answers
30 views
The sum of the reciprocals of fourth powers [duplicate]
This problem is an extension of the well known basel problem and involves finding the sum of
1 + 1/16 + 1/81 ... = 1/1^4 + 1/2^4 + 1/3^4 ... 1/n^4 where n tends to infinity
Euler managed to prove ...
5
votes
1answer
66 views
finding $\lim_{n\rightarrow\infty}\sum_{r=1}^{n}\arctan\left(\frac{2r}{1-r^2+r^4}\right)$
I want to find
$$L=\lim_{n\rightarrow\infty}\sum_{r=1}^{n}\arctan\left(\frac{2r}{1-r^2+r^4}\right)$$
I already know that I need to split the expression $\frac{2r}{1-r^2+r^4}$ of the form ...
31
votes
5answers
894 views
How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$?
How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$
I found the above interesting identity in the book $\bf \pi$ Unleashed.
Does anyone knows how to ...
3
votes
2answers
75 views
Please, help me to find where is a mistake in the solutions of the equation.
I have this equation and I will be very thankful to anyone who can provide me any help with the one discrepancy in my solution and the solution from the self-learning website:
$$
\frac{1+\tan(x) + ...
0
votes
1answer
78 views
Sines and cosines of angles in arithmetic progression
Prove that if $\phi$ is not equal to $2k\pi$ for any integer $k$, then
$$\sum_{t=0}^{n} \sin{(\theta + t \phi)}=\frac{\sin({\frac{(n+1)\phi}2})\sin{(\theta+\frac{n \phi}2)}}{\sin{(\frac{\phi}2)}}$$
...
8
votes
2answers
175 views
How to find finite trigonometric products
I wonder how to prove ?
$$\prod_{k=1}^{n}\left(1+2\cos\frac{2\pi 3^k}{3^n+1} \right)=1$$
give me a tip
8
votes
1answer
1k views
A hard definite integral with trignometric
How could we get a closed form for this one?
$$\displaystyle\int_{0}^{\frac{\pi }{2}}{{{x}^{2}}\sqrt{\tan x}\sin \left( 2x \right)\text{d}x}$$
1
vote
1answer
100 views
Simplifying trigonometric summation: $ x[n] = \sum_{k = 1}^{\infty} (-1)^{k} \frac{\sin \left( 2 \pi k M \frac{a}{b} n \right)}{\pi k}$
I was reading an engineering publication and attempting to follow the math and got stuck at this "easy to show but somewhat lengthy" step. The author starts with
$$
x[n] = \sum_{k = 1}^{\infty} ...
10
votes
3answers
336 views
$\tan(x) = x$. Find the values of $x$
How can I find the possible values of $x$ for:
$\tan(x)=x$
mathematically?
3
votes
1answer
444 views
Finding the sum of a trigonometric series
Find the sum of the series $$\cos x + \cos 2x + \cdots + \cos (n-1)x.$$
You must calculate the sum of this series only by multiplying through by $2\sin\left(\frac{x}{2}\right)$.
Now I've heard of ...
4
votes
1answer
102 views
Showing that $ \sum_{n=1}^{\infty} \arctan \left( \frac{2}{n^2} \right) =\frac{3\pi}{4}$
I would like to show that:
$$ \sum_{n=1}^{\infty} \arctan \left( \frac{2}{n^2} \right) =\frac{3\pi}{4}$$
We have:
$$ \sum_{n=1}^N \arctan \left( \frac{2}{n^2} \right) =\sum_{n=1}^N \arctan ...
1
vote
1answer
111 views
Cauchy product and the exponential function
Simplify the following series using the Cauchy product
$$\sum\limits_{k=1}^\infty\frac{1}{k!}\cdot\sum\limits_{j=1}^\infty\frac{1}{j!}$$
...
2
votes
0answers
46 views
tangents of infinite sums — reference request
This section of Wikipedia's List of trigonometric identities was, as far as I know, written entirely by me. For a time, it dealt only with finite sums, and said something like this:
$$
...
3
votes
1answer
75 views
A curious identity on sums of secants
I was working on proving a variant of Markov's inequality, and in doing so I managed to come across an interesting (conjectured) identity for any $n\in\mathbb{N}$:
$$\sum_{m=0}^{n-1} ...
3
votes
5answers
621 views
Sequence of solutions to $x\sin x=1$
Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed.
Consider a sequence $x_n, n\ge1$ formed by positive solutions to ...
9
votes
3answers
454 views
Explicitly finding the sum of $\arctan(1/(n^2+n+1))$
This is from a GRE prep book, so I know the solution and process but I thought it was an interesting question: Explicitly evaluate $$\sum_{n=1}^{m}\arctan\left({\frac{1}{{n^2+n+1}}}\right)$$.
7
votes
1answer
173 views
Help in evaluating $\sum\limits_{k=1}^\infty \frac1{k}\sin (\frac{a}{k})$
I would like to try to evaluate
$$\sum\limits_{k=1}^\infty \frac{\sin (\frac{a}{k})}{k}$$
However, all of my attempts have been fruitless. Even Wolfram Alpha cannot evaluate this sum. Can someone ...
2
votes
2answers
204 views
Showing $f(x)=\sum_{n=1}^{\infty}{\sin\left(\frac{x}{n^2}\right)}$ is continuous.
Let
$$f(x)=\sum_{n=1}^{\infty}{\sin\left(\frac{x}{n^2}\right)}.$$
a) Show that the series converges for $x\in [0,\pi/2]$.
b) Show that $f$ is monotone and continuous on this interval.
This is what ...
1
vote
1answer
196 views
Use the identity $\cos(A-B) -\cos(A+B) = 2\sin(A)\sin(B)$ to prove another identity and evaluate a sum
Use the identity $$\cos(A-B) -\cos(A+B) = 2\sin(A)\sin(B)$$ to prove that:
$$2\sin(j\theta)\sin(\frac{1}{2}\theta)=\cos((j-\frac{1}{2})(\theta))-\cos((j+\frac{1}{2})\theta).$$
This seemed almost too ...
5
votes
1answer
140 views
tangents of sums
In the identity
$$
\cos \left( \sum_i \theta_i \right) = \sum_{\text{even }n\ge0} (-1)^{n/2} \sum_{|I|=n} \prod_{i\in I} \sin\theta_i \prod_{i\not\in I}\cos\theta_i
$$
one can prove the case of ...
4
votes
3answers
327 views
Partial Fractions Expansion of $\tanh(z)/z$
I have seen the following formula in papers (without citations) and in Mathematica's documentation about Tanh[]:
$$
\frac{\tanh(z)}{8z}=\sum_{k=1}^{\infty} ...
4
votes
1answer
482 views
Show $1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}}$ for $x \neq 0$
For $x \neq 0$, $$ 1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}} $$
19
votes
2answers
400 views
an integer sum of products of tangents
This question arose from my initial attempts at answering this question. I later found a way to transform the desired sum into a sum of squares of tangents, but before I did, I found numerically that ...
30
votes
4answers
541 views
Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$
Prove that $$\sum _{l=1}^{n}\sum _{k=1}^{n-1}\tan \frac {lk\pi } {2n+1}\tan \frac {l( k+1) \pi } {2n+1}=0$$
It is very easy to prove this identity for each fixed $n$ . For example let $n = 6$; ...
9
votes
4answers
617 views
Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$
Recently, I ran across a product that seems interesting.
Does anyone know how to get to the closed form:
$$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)=-\frac{\sin(\frac{n\pi}{2})}{2^{n-1}}$$
I ...
4
votes
2answers
308 views
How do we know Taylor's Series works with complex numbers?
Euler famously used the Taylor's Series of $\exp$:
$$\exp (x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$
and made the substitution $x=i\theta$ to find
$$\exp(i\theta) = \cos (\theta)+i\sin (\theta)$$
How ...
10
votes
1answer
364 views
For a trigonometric polynomial $P$, can $\lim \limits_{n \to \infty} P(n^2) = 0$ without $P(n^2) = 0$?
Disclaimer: The original version of this question focused on $2^n$ in lieu of $n^2$. It is in the hope that the question is easier with $n^2$ that I changed it.
I have an always-nonnegative (on the ...
6
votes
2answers
175 views
Calculate: $\sum_{k=0}^{n-2} 2^{k} \tan \left(\frac{\pi}{2^{n-k}}\right)$
Calculate the following sum for integers $n\ge2$:
$$\sum_{k=0}^{n-2} 2^{k} \tan \left(\frac{\pi}{2^{n-k}}\right)$$
I'm trying to obtain a closed form if that is possible.
4
votes
2answers
300 views
Proof about $z\cot z=1-2\sum_{k\ge1}z^2/(k^2\pi^2-z^2)$
In Concrete Mathematics, it is said that
$$z\cot z=1-2\sum_{k\ge1}\frac{z^2}{k^2\pi^2-z^2}\tag1$$
and proved in EXERCISE 6.73
$$z\cot z=\frac z{2^n}\cot\frac z{2^n}-\frac z{2^n}\tan\frac ...
1
vote
1answer
87 views
Sum Cosine Mod?
interval $-\pi:\pi$ split into M equal intervals.
midpoints are $y_K$
but i dont understand how to show
$$
\frac{1}{M}\sum_{j=1}^{M}\cos(mx_{j})=\begin{cases} 1, & \ m \equiv 0\pmod{M}\\ 0, ...
8
votes
1answer
365 views
Infinite product of sine function
How to prove the following product?
$$\frac{\sin(x)}{x}=
\left(1+\frac{x}{\pi}\right)
\left(1-\frac{x}{\pi}\right)
\left(1+\frac{x}{2\pi}\right)
\left(1-\frac{x}{2\pi}\right)
...
0
votes
1answer
148 views
Trig Series Sum
I have equation that I try to solve for one of the values
$$\sum_{k=0}^{n-1}\cos(2 \pi fk)(x_{k}- \mu-A\cos(2 \pi fk)-B\sin(2 \pi fk))$$
I know to set equal $0$
I try to solve for A but how to take ...
1
vote
0answers
39 views
3
votes
2answers
146 views
Finding $\tan t$ if $t=\sum\tan^{-1}(1/2t^2)$
I am solving this problem. If $$\sum\limits_{i=1}^{\infty} \tan^{-1}\biggl(\frac{1}{2i^{2}}\biggr)= t$$ then find the value of $\tan{t}$.
My solution is like the following: I can rewrite:
...
2
votes
2answers
163 views
Show that $\sum\limits_{i=1}^N\sin^2\frac{m\pi i}{N+1}=\frac{N+1}{2}$
I have recently found this excercise and was not able to solve it so far.
Show that
$$\sum_{i=1}^N\sin^2\frac{m\pi i}{N+1}=\frac{N+1}{2}\;,$$ where $m \in \lbrace1,2,...,N\rbrace$.
This was one of ...
7
votes
1answer
376 views
How to turn this sum into an integral?
I have been trying to find the closed form of this sum to no avail. It was suggested to me to try and turn this sum into an integral and solve it like that. However, I am confused as to how to do ...
3
votes
2answers
121 views
Determine convergence of $\sum_{n=1}^{\infty} (\cos{\frac{2}{n}}-\cos{\frac{4}{n}})$
Determine convergence of $$\sum_{n=1}^{\infty} \left(\cos{\frac{2}{n}}-\cos{\frac{4}{n}}\right)$$
In the answer, it says
$$\cos{\frac{2}{n}}-\cos{\frac{4}{n}} = 2\sin{\frac{3}{n}}\sin{\frac{1}{n}} ...
6
votes
1answer
358 views
sum of series involving coth using complex analysis
I am self-studying complex analysis, so I am a rookie. I ran across an interesting series I am trying to evaluate using CA.
Show that $$\sum_{n=1}^{\infty}\frac{\coth(\pi ...
