Mathematical Equality

[m_the0ry]

Lets say x is a real number

nearest(x)= x + f

Where f and x are both real numbers but their sum is an integer. For example

nearest(2.32) = 2.32 - .32
nearest(2.45) = 2.45 - .45
nearest(2.62) = 2.62 + .38

f linearly increases with x, forming a ramp of slope 1 spanning from -.5 to .5 over an interval of 1. At that point it becomes discontinuous and starts over again. This is obviously a periodic function and thus is very well described by a fourier series. Going from periodic function to fourier series can be tricky so I cheated, referencing the common 'saw wave' signal which has the exact same form except translated in a few ways. I just modified that function and then derived the rest.

One thing I noted is that technically the saw function in fourier form is defined at .5 as being zero. But if you infinitesimally translate the function to the left the sinusoidal series' still converge to zero.

EDIT: I made a mistake in the first post the sinusoids only converge to zero with respect to i.