fuckyeahphysica:

Fibonacci sequence in the hiding…

Proof (Beautiful) :

The only thing that you need to know to prove this is that if the Fibonacci numbers were the coefficients to a power series expansion, then the Fibonacci generating function is given as follows:

Substituting the value of x = 1/10, we get :

Proved.

Have a  great day!

Beautiful Proofs(#3): Area under a sine curve !

So, I read this post on the the area of the sine curve some time ago and in the bottom was this equally amazing comment :

Screenshot from 2017-06-07 00:19:11

\sum sin(\theta)d\theta =   Diameter of the circle/ The distance covered along the x axis starting from 0 and ending up at \pi.

And therefore by the same logic, it is extremely intuitive to see why:

\int\limits_{0}^{2\pi} sin/cos(x) dx = 0

Because if a dude starts at 0 and ends at 0/ 2\pi/ 4\pi \hdots, the effective distance that he covers is 0.

Circle_cos_sin.gif

If you still have trouble understanding, follow the blue point in the above gif and hopefully things become clearer.

 

Beautiful proofs (#1) : Divergence of the harmonic series

The harmonic series are as follows:

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And it has been known since as early as 1350 that this series diverges. Oresme’s proof to it is just so beautiful.

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Now replace ever term in the bracket with the lowest term that is present in it. This will give a lower bound on S1.

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Clearly the lower bound of S1 diverges and therefore S1 also diverges.
But it interesting to note that of divergence is incredibly small: 10 billion terms in the series only adds up to around 23.6 !

Divisibility tests exposed!

Number theory is the home to a
myriad of beautiful proofs, and let’s set forth on this journey by
learning about divisibility tests.

Have a great day!

Yet another day at the board !

This time its the geometric series formula. The simplicity and elegance of the derivation inspired me to post this here.

Have a great day!

All numbers are Interesting

Proof:

Suppose not, then there exists a smallest
non-interesting number.

But being the smallest non-interesting number singles it out and
thus making it an interesting number. Hence a contradiction.

All numbers are Boring

Proof: 

Suppose not, then there exists a smallest non-boring number.

But being the smallest non-boring number singles it out and
thus making it a boring number. Hence a contradiction.

What have i gotten myself into !

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