/sci/ - Science & Math

let <span class="math">J_n = \int_0^{\pi/2} \frac{sin((2n+1)x}{sin(x)} dx[/spoiler]

and <span class="math">K_n = \int_0^{\pi/2} \frac{sin((2n+1)x}{x} dx [/spoiler]

show that <span class="math">J_n - K_n \rightarrow 0 [/spoiler], that <span class="math">\forall n, J_n = \frac{\pi}{2}[/spoiler], and that <span class="math">K_n \rightarrow \int_0^{\infty} \frac{sin(x)}[x}dx[/spoiler].

>>5861824
well I don't know how you got there either, but I'm sure it's not possible since your expression has a value of 1 while all the terms of the product have a modulus of less than 1!

>>5861816
I'll be going to sleep in one hour or two but I'll make sure to check the thread tomorrow first thing in the morning!

>>5861798
Use <span class="math">cos(a)cos(b)= \frac{cos(a-b)+cos(a+b)}{2}[/spoiler] with good values of a and b!

>>5861724
try expressing the whole thing in terms of cos(pi/3) instead ;)

bump, is anyone lurking or trying other than >>5861222 ?

>>5861222
the equation is equivalent to <span class="math">sin(x)-xcos(x)=0[/spoiler].
From there, you could have some intuition by using the first terms of sin and cos series!

bump!
post any partial solution or any idea you have!

I'll keep lurking in case anyone needs tips or has any question.
I hope someone likes any of those!

and the final one is for other undergrads who are more confident:

consider the increasing sequence <span class="math">\displaystyle (\lambda_{n})_{n\in \mathbb{N}}[/spoiler] of the positive real solutions of the equation <span class="math">\displaystyle \tan(x)=x[/spoiler].

Show that <span class="math">\displaystyle \sum_{n=1}^{\infty} \frac{1}{\lambda_{n}^{2}}=\frac{1}{10}[/spoiler]

the second one is most likely for undergrads, in the early stages:
study all sequences such as <span class="math">u_{0}\ge0[/spoiler] and <span class="math">u_{n+1}=\frac{1}{2}(u_{n}^{2}+u_{n})[/spoiler].

Then, if it exists, find the limit of <span class="math">(u_{n}^{1/n})[/spoiler] depending on <span class="math">u_0[/spoiler].

Hello.
Here is my new attempt at posting gradual problems, for people of different levels so that you can try them for fun.
The first problem can be done using usual trig formulas and can be done after highschool:
show that <span class="math">cos(\frac{\pi}{9})cos(\frac{2\pi}{9})cos(\frac{3\pi}{9})cos(\frac{4\pi}{9}) = \frac{1}{16}[/spoiler]

>>5858715
>>5858724
should I give hints?

for example, in the first problem, you have to show that the line of equation y=x always intersects the line of equation y=f(x) at only one point.
You could study an intermediary function...
pic related

>>5858715
yet if you choose x=y=z=1/2 it works for f(x)=-x+1!

I should add that f is continuous.
This thread is fucked isn't it?

>>5858673
the third problem is asking for an equivalent as n goes to infinity!

>>5858661
first, thank you for your appreciation, I thought this could be a good idea!
Second:
The notation means <span class="math"> a < c < b [/spoiler] (I don't even know why I didn't write it this way in the first place)

>>5858673
P is a polynomial, sorry
I was so affraid the latex wouldn't work I didn't pay attention to those details!

>>5858628
superman would too!

hey /sci/

today I thought we could use some problems at different levels, problems which you could try your hand on.
If you like it, I'll post more topics like this in the future stating:
-an approximate level at which you can hope to find a solution
-the problem itself

First problem, which you can probably do after highschool:
let <span class="math">f : \mathbb{R} \rightarrow \mathbb{R}[/spoiler] be a strictly decreasing function.
Show that the system <span class="math"> \left\{ x=f(y), y=f(z), z=f(x) \right\} [/spoiler] has a unique solution.

Second problem: freshmen can try it!
let <span class="math">a<b \in \mathbb{R}[/spoiler]. Show that there exists a unique <span class="math">c \in ]a,b[ [/spoiler] such as <span class="math"> \int_{a}^{b} P(x)dx[/spoiler] is a linear combination of <span class="math">P(a), P(b)[/spoiler] and <span class="math">P(c)[/spoiler].

Third problem is for people with more experience I think:
Give an equivalent of <span class="math"> \displaystyle \sum_{k=1}^nk \lfloor \frac{n}{k} \rfloor [/spoiler]

There you go, good luck and ask anything!

hey /sci/

today I thought we could use some problems at different levels, problems which you could try your hand on.
If you like it, I'll post more topics like this in the future stating:
-an approximate level at which you can hope to find a solution
-the problem itself

First problem, which you can probably do after highschool:
let <span class="math">f : \mathbb{R} \rightarrow \mathbb{R}[/spoiler] be a strictly decreasing function.
Show that the system <span class="math"> \left\{ x=f(y), y=f(z), z=f(x) \right\} [/spoiler] has a unique solution.

Second problem: freshmen can try it!
let <span class="math">a<b \in \mathbb{R}[/spoiler]. Show that there exists a unique <span class="math">c \in ]a,b[ [/spoiler] such as <span class="math"> \int_{a}^{b} P(x)dx[/spoiler] is a linear combination of <span class="math">P(a), P(b)[/spoiler] and <span class="math">P(c)[/spoiler].

Third problem is for people with more experience I think:
Give an equivalent of <span class="math"> \displaystyle\sum_{k=1}^nkE\left(\frac{n}{k}\right) [/spoiler]

There you go, good luck and ask anything!

The basic principle is to use an astable multivibrator to create a square signal
http://en.wikipedia.org/wiki/Multivibrator#Astable_multivibrator

and then simply use an integrator to create the desired triangular sawtooth waveform.

there are two major characterizations:

the first one is that F is nilpotent iff X^n is the characteristic polynomial of F.
This is obviously not the one we're gonna use.

The other one is that F is nilpotent iff for all p between 1 and n, tr(F^p)=0

This one is easy to apply here using the basis B (the trace doesn't depend on the basis).

Shall I let you try?

>>5740800
I've just checked it out, P is indeed at a distance sqrt(2) of the plane S: x+z=2, and S contains your line.

>>5740760

first, find the distance of P to the line, you find <span class="math">\sqrt{2}[/spoiler] if I'm not mistaken.
Which means that in fact the line is at the same distance to P as the place is.

You can then easily find the point on the line that minimizes the distance to P:
I called it H, and I find <span class="math">H ( 2,1,0)[/spoiler].

You now know that <span class="math">\vec{HP}[/spoiler] is orthogonal to your plane (because H is the orthogonal projection of P on the plane), so you found an equation of the plane:
<span class="math"> x+z = 2 [/spoiler]

(I hope I didn't make any mistake)