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/sci/ - Science & Math

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5861121 No.5861121 [Reply] [Original]

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]

>> No.5861128

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].

>> No.5861136

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]

>> No.5861158

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

>> No.5861214

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

>> No.5861222

>>5861136

Saw something like this on a Math GRE one time. Could you give us a hint?

>> No.5861233

>>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!

>> No.5861341

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

>> No.5861504

this is getting sad

>> No.5861592

/sci/ you're hopeless

>> No.5861614

>>5861121

/sci/ is for pop sci, religion debates, major meta threads, highschool and calculus 1 hw, and other random questions.

while threads that pose any challenging questions are deemed as hw and everyone tries to get the OP banned.

>> No.5861724

Starting the first one, I'm thinking of writing it all out in multiple angle formulae in terms of cos(pi/9) but I'm thinking that's wrong.

>> No.5861766

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

>> No.5861778

Basically a hanndless pasta strainer meets a hand screw bottle opener.... & that shits all superpositioned

I'm sorry OP but when the formioli doesn't have a purpose its pretty weird to write a proof

>> No.5861798

>>5861766
That sounds like it would be nicer but I'm not sure what an identity for cos(A/3) would be

A being pi/3

>> No.5861811

>>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!

>> No.5861816

>>5861811
agh this is getting annoying but I want to solve it now

will be back at some point if the thread is still up

>> No.5861823

>>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!

>> No.5861824

On the first one I made it to
1/4[2cos(π)cos(-2π)][2cos(3π)cos(-4π)]
but that's probably wrong. I have never even seen some of the trig formulas I used for it, and I've done pre-calc 12.

>> No.5861838

>>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!