/sci/ - Science & Math

After the first challenge got ruined by some ass holes, I decided to make another.
Prove that "<span class="math">\displaystyle\cot(x)=\frac{1}{x}-\sum\limits_{k=1}^\infty\left(\frac{2x}{(\pi \cdot k)^2-x^2)}\right)[/spoiler]" is true.

The problem requires very basic Math but is very challenging nonetheless. I proved the formula by accident but doing on purpose would be much harder.

Since it is hard, I will keep on adding hints and they will all be as a reply to the OP.

1st Hint: You'll need Euler's infinite product for the Sine function.

Little Mathematical challenge for those of you interested:
Prove that "<span class="math">\displaystyle\sum\limits_{k=1}^{\infty}\left(\displaystyle\frac{1}{k \cdot x^k}\right)=\ln\left(\displaystyle\frac{x}{x-1}\right)[/spoiler]" is true for any real "<span class="math">x[/spoiler]" greater than 1.
Good luck and I will post my solution when a few of you post yours.

Little Mathematical challenge for those of you interested:
Prove that "\displaystyle\sum\limits_{k=1}^{\infty}\left(\displaystyle\frac{1}{k \cdot x^k}\right)=\ln\left(\displaystyle\frac{x}{x-1}\right)" is true for any real "<span class="math">x[/spoiler]" greater than 1.
Good luck and I will post my solution when a few of you post yours.

Doesn't it blow your fucking mind that the sum <span class="math">\displaystyle\sum\limits_{k=1}^{\infty}\left(\displaystyle\frac{1}{P_k}\right)[/spoiler], where <span class="math">P_k[/spoiler] is the kth prime, diverges but <span class="math">\displaystyle\sum\limits_{k=1}^{\infty}\left(\displaystyle\frac{1}{k^{1+\epsilon}}\right)[/spoiler] converges for ANY <span class="math">\epsilon>0[/spoiler], no matter how small of an epsilon.
Topic of thread: Post mathematical "stuff" that still blow your mind even after you proved them
and understood them.

Doesn't it blow your fucking mind that the sum <span class="math">\displaystyle\sum\limits_{k=1}^{\infty}\left(\displaystyle\frac{1}{P_k}\right[/spoiler], where <span class="math">P_k[/spoiler] is the kth prime, diverges but <span class="math">\displaystyle\sum\limits_{k=1}^{\infty}\left(\displaystyle\frac{1}{k^{1+\epsilon}}\right)[/spoiler] converges for ANY <span class="math">\epsilon>0[/spoiler], no matter how small of an epsilon.
Topic of thread: Post mathematical "stuff" that still blow your mind even after you proved them
and understood them.

Computer Science
Liberal

Society goes through many zeitgeists and if you stick with one affiliation for far too long, you will look like an ass..just like the Republicunts of 2012.

bump pls

Is there current research into cognitive augmentation for already born human beings with means other than genetical ones.
If so, can you please share them?

Subject of the thread is simple;
Share the websites you use to download ebooks.
Discover new cool websites to find what you want and share websites that you think deserve more "fame".

I'll start:
en.bookfi.org

It's got a huge library and is based in Russia.

>>5245783
beautiful, many thanks anon.

http://www.khanacademy.org/ great step by step videos on whatver you need. make an account and build your knowledge!

“If the human brain were so simple that we could understand it, we would be so simple that we couldn’t.”
- Emerson M. Pugh

There's something more at play here...

I just got a call from an alarm company notifying me that there's a technical problem with my alarm.
Thought this was strange as I don't have an alarm. I googled the address they gave me and the issue came from the Tandem Van De Graaff/Cyclotron at the Brookhaven National Laboratory.
I just got this phone, and now I'd really like to know who had this number before me.

hey guys, i was just wondering.
imagine something like that:

you want to learn some math;
these are the rules:

you can only see the book for 1 minute/day.
what would you do?

imagine it. you would run straight forward to the definition of stuff, right?

than, the rest of the day, you will not be able to learn "more" from the book.

you will have a whole day just to think and abstractize and imagine the definition. what it meant.

you will be taking a shower and thinking of it and its implications.

then, the next day, you will see the book and "omg these exercices are so easy"
"these are useless, these things are obvious and easily taken from the definition"

yes, maybe, no?

Solve the following quadratic equation using the most efficient technique (factoring, extracting, square roots, completing the square, or using the quadratic formula).

2(4x-9)^2+5=69

Let vector A=(1,2) Determine matrix [perp onto a]
I dont even know where to start anything to get me started would be appreciated.

I was playing around with the serie <span class="math">\sum_{k=0}^n k^2[/spoiler] and came up with the interesting identity <span class="math">\sum_{k=0}^{2n} k^2 = 2 n^2 \left ( n+\frac{1}{2} \right )+2 \sum_{k=0}^n k^2[/spoiler] (I can show you how I did it but it will take some time to write everything even though it's very easy)... So I wondered if I could make an approximation of <span class="math">\sum_{k=0}^{2n} k^2[/spoiler] from this by substituting <span class="math">2*\sum_{k=0}^{n} k^2[/spoiler] every time and of course you can.. So I did it and there are two approximations I came up with <span class="math">\sum_{k=0}^{2n} k^2=\sum_{k=0}^{\infty}\left ( 2^{k+1}\left ( \frac{n}{2^k} \right )\left ( \frac{n}{2^k}+\frac{1}{2} \right ) \right ) AND \sum_{k=0}^{2n} k^2=\sum_{k=0}^{n}\left ( 2^{k+1}\left ( \frac{n}{2^k} \right )\left ( \frac{n}{2^k}+\frac{1}{2} \right ) \right )+1[/spoiler] and as I expected <span class="math">\sum_{k=0}^{2n} k^2=\sum_{k=0}^{n}\left ( 2^{k+1}\left ( \frac{n}{2^k} \right )\left ( \frac{n}{2^k}+\frac{1}{2} \right ) \right )+1[/spoiler] was much more close to the real answer by the other equation seemed to have more in common with the equation resulting from <span class="math">\sum_{k=0}^{2n}\left (k^{2}\right )[/spoiler] which is <span class="math">\sum_{k=0}^{2n}\left (k^{2}\right )= \frac{1}{3} n \left (2 n+1\right ) \left (4 n+1\right )[/spoiler]

/Continuation in the next post because field is too big to post

Hello /sci/ I need some help(big surprise im sure), I love math, but haven't touched it in years due to work/life and finally have some free time again to start learning all over again, i got thru advanced calc in highschool and took a few math theory classes in uni, but haven't used any or even looked at it in about 6 or 7 years, where would be the best place for me to start again, Trig, Algebra, and geomentry are all fine Didn't lose anything there, just wondering what a good general path to start down the road again is.

rate this algorithm to find arctg

function arctg(t, dtheta)
{ for (theta=0; tan(theta) < t; theta += dtheta) ;
return theta;
}

How do you get from cos(2x)-sin(2x)tan(a)

to

Sec(a)Cos(a+2x)

I can't figure it out for some reason, thanks for the help!

Somebody please link me to that websites that archives everything on /sci/ (And two other boards, /lit/ or /g/ being one of them, I think). It has every single post. Deleted and non deleted, old and new, ALL!
Thanks in advance. I will delete this thread in the same 15 seconds that I will get the answer.

Use the Intermediate Value Theorem to show that the equation
x^3+ 10x^2+ 8x − 50 = 0
has roots between 1 and 2, between -4 and -3 and between -9 and -8.

penguins.

Solve the following pair of simultaneous equations. What is the value of X?
10Y=3X+2
2Y=2X-1

Help me pleas