The Magic of Mathematics – Adding up all natural numbers = negative

Came across this and was just amazed, having studied mathematics at university it’s hard to believe that the following statement could be true

\sum\limits_{n=1}^\infty n = - \frac{1}{12}.

However, check out this “proof” to show that it is.

Lets take

S_1 = 1-1+1-1+1-1+1... = \frac{1}{2} .

S_2 = 1-2+3-4+5-6+....

S = 1+2+3+4+5+6....

Now, 2S_2 can be written as

  1-2+3-4+5-6+…
+   1-2+3-4+5-6+…
= 1-1+1-1+1-1+1-…
= S_1.

So we can deduce that 2S_2 = S_1. So it follows that S_2 = \frac{1}{2} S_1=\frac{1}{4}.

Now lets take S - S_2 = 1+2+3+4+5+... - (1-2+3-4+5-...) = 4+8+12+16+... = 4S.

It follows that S - \frac{1}{4} = 4S.

Rearranging gives -\frac{1}{4} = 3S.

Giving S= - \frac{1}{12}.

Now, S = 1+2+3+4+5+....

Therefore,  \sum\limits_{n=1}^\infty n = - \frac{1}{12}. Right?

I found this result absolutely amazing – but this result is used in string theory and quantum mechanics!

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Choas Theory: Hénon Map (Part 3)

       Unstable periodic orbits

In this section we’ll be introducing a method to calculate the unstable periodic orbits (UPO’s) of the Hénon map, showing that by plotting them we can create the Hénon attractor. By using equation (5) and solving the differential (6) then we can find the UPO’s and plot them [BW89].

F_n = -x_{n+1}+a-x_n^2+bx_{n-1}, \qquad \qquad \qquad \qquad (5)

dx_n/dt = s_nF_n, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \, \, \, \, (6)

where s_n = \pm1 and n=1,...,p. Using MATLAB we can solve this differential and find the UPO’s of order n for the Hénon map. Since we’re introducing the method we will look up to period 6. The Figure below shows the UPO’s for n=1, n=2, n=4 and n=6, note that when n=3 or n=5 the orbits are non-existent.

a8c2bb9303e1ad17b96b54a86bfc8428.epsUnstable periodic orbits of order n

By using MATLAB coding we can compute the fixed points that lie on the periodic orbits.

henon_UPOsHénon attractor with points of UPO’s denoted by colours black, red, yellow and green

The figure above shows that for the first 6 periodic orbits, the fixed points lie on the attractor, if we continue this process up to period n and plotted the orbits we will be able to form the Hénon attractor. The black point denotes the period-1 UPO, red period-2, yellow period-4 and green period-6. Note that there exists more periodic orbits and fixed points up to period-6, i.e. there are two period-6 orbits.

Sources:

[BW89] WENZEL, B. (1989). PHYSICAL REVIEW LETTERS. Volume 63, Number 8, p.819-822. Ohio State University, USA.