M2023 - Fourier Series : 3 - arith. freq + geo. amp =? circle

by using fourier series, we can create a lot of periodic waves. here are some of the famous, important waves for examples. In this case, although the final trace on the 2D complex plane doesn't resemble any waves, but we can keep track of the final Y-position of the final vector v[t]

sawtooth :

square :

here the amplitudes are decreasing as the denominator is arithmatically increasing.

but what if the amplitude is decreaseing as the denominator is geomatrically increasing ?

in other words, here's an example :

if we don't care about the green trail (Y-position) for now, we can see that the final white trail v[t] traces ... a circle ?

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I tried with other shrinking ratios, and we ended up with different circles.

Let's rewrite the whole thing in a more math friendlier way.

this means that the final vector v[t] should look something like this :

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first, find v[0], then find v[π]

these 2 are the "right most" and "left most" of the "circle" //let's assume it is a circle first

then we can find the radius : ( v[0] - v[π] ) / 2 = r

then the center point : v[0] - r = c

the final big part is to proof that the distance between v[t] and c will stay constant

this proof may take a long while to finish, maybe months, maybe years.

here we go

this is what I have so far

we are not done yet