MRB constant O

Let m = the MRB constant = = 0.1878596...

Proof that

To simplify the appearance of our work a little, below an infinite series of  is what is symbolized by .

Let m= the sum of the convergent infinite series of

In general we have the following:

Expanding

Collecting the constants we see that

By Grandi's series we know that

Collecting the infinite series we see that

Specifically, here where z = 1, by the lema

by one of the commutative laws the Cesaro sum of the divergent infinite series of 

Since

 (-1)^n = and  and the Cesaro sum of the divergent infinite series of

Also since and the Cesaro sum of the divergent infinite series of .

Let t=, and then the infinite series of  = infinite series of exp(I*t) = .

Which means that 

Now we have t=, and since n>0

we can expand ln(n) into the series .

Thus the expanded version of t is t=.

Putting the expanded version of t into the formula  gives =m.

Distributing the "I" we get

Since  = , we have

Distributing the 1/n we get

=

Notice that "I" and the innermost series are both raised to the "k"th power, so we can combine like powers and get

Obviously -(I^2) =1; so we have

Moving the innermost series to the left gives us

Therefore distributing the 1/k! we get

QED