Euler's sum : Software simulation for 1/1²+1/2²+1/3²+1/4²+ ⋯ = π² / 6

Hi,

Very famous Euler's sum must be radian !

 1/1²+1/2²+1/3² + ... = 1/1 + 1/4 + 1/9 + . . .                 Equation (1)

What if the start term ( 1 over 1 ) is interpreted as arc-length 1 over circle radius 1 in view of geometric ? So the angle value of 1 over 1 is 1 radian or 57.29 degree. The second, the third can be calculated as 0.25(=14.32 degree), 0.111(=6.35 degree).

All degree can be summed up so we can show these accumulated sum as graphically.

Another words, all sequential number turns around circle as we sum up to infinity. Why not ? If this is proved right, that would be unbelievable but could be true !

There is detail of this below.

My simulation's result is like below.

There are some notes for above picture.

1) There are red, cyan, yellow arc and green filled arcs in the graphics.

As every term in Equation (1) continues to increase, the color change from red to green and then red again circulately. So, there are color sequence like 1st(red), 2nd(cyan), 3rd(yellow), 4th(green), 5th(red), 6th(cyan), 7th(yellow), 8th(green), 9th(red),  and so on.

2) Each arc shows that in example 1/1² = 1/1 = (arc-length) / (radius),  1/2² = 1/4 = (arc-length) / (radius), and so on.

3) The left side's white number is 94.24777 in degree same as π²/6 radian.

4) The left side's colored number ( changing from red to yellow and circular way ) is the latest calculated sum as being simulated.

We can think the case that you sum up to 3rd one,  that is, 1/1²+1/2²+1/3² = 1+0.25+0.111 = 1.361 radian ( 77.985 in degree ) and there shows 1st(red), 2nd(cyan) and 3rd(yellow) filled-arc in graphic. See next picture.

Now, here is total simulation.

If you see this video capture of my math simulation program of Euler's sum, you would know why I matched 1 over n^2 ( 1, 4, 9, . . . ) as arc-length over radius of circle.

The arcs' sum-up degree is approaching to 94.24777 in degree same as π²/6 radian.

Amazing !

From this simulation, I felt that sequential numbers also turn around ther center of circle like that planets turns around Sun. (^^) Normally we think sequence of number is aligned straight line. But acually, we walk arc way in Earth with center of Earth if you walk long long way in example asia to america. (^^)

After you see this video, you may think other sequencial sum.

Like other riemann zeta functions or geometric sum, etc.

Above simulation is the case of s=2 as Riemann zeta function's input.

I want to make other my creative blog about 1+1+1+ ... = -1/2 with this concept in near future. So this would be very interesting math simulation but needs more time to make sure that it is right and need to make tangible one like above simulation.

I prepared this concept for long time.

Finally write and upload this. 

Especially in memory of death my uncle, Kim Ki Young, that I love most.

Thanks for read.