Berg's Theorem

formerly Howell's Conjecture

proof c. 1997

Most curious students of arithmetic notice the interesting sequence of digits that results from inverses like 1/98 = 0.010204081632.... Here we prove the source of these "coincidences" and show that they apply to a wide class of rational numbers whose patterns are not so immediately obvious.

1 / (10^p - d) = SUM(k=0 to inf, d^k / (10 ^ p(k+1) ))

In the example above, p ("power of 10") is 2, and d ("distance from 10^p") is 2.

If we substitute x = d, y = 10^p, the LHS is a function of the form (y - x)^-1 = g(x).

There is a Taylor polynomial f(x) that approaches g(x) with arbitrary precision, where f(0) = g(0), f'(0) = g'(0), and so on. In particular:
g(0) = 1/y = f(0)
g'(0) = 1!/y^2 = f'(0)
g''(0) = 2!/y^3 = f''(0)
-- etc. --

If we write f(x) = c0 + c1*x + c2*x^2 + ... then,
c0 = f(0) => c0 = 1/y
c1 = f'(0) => c1 = 1/y^2
2!c2 = f''(0) => c2 = 1/y^3
-- etc. --

So, f(x) = 1/y + x/y^2 + x^2/y^3 ... or after resubstitution,
= 1/10^p + d/10^2p + d^2/10^3p ...
= SUM(k=0 to inf, d^k / (10 ^ p(k+1) )), QED

This sum converges when LIM(n->inf, abs( A_n+1 / A_n )) < 1
L = lim(k->inf, d^(k+1) / 10^p(k+2) * 10^p(k+1) / d^k)
L = lim(k->inf, d / 10^p)
L = d / 10^p
If L < 1, then 10^p > d [since 10^p > 0]

Thus, the sum converges where d < 10^p [which does not in fact limit the inverses to which the theorem applies so long as p and d are chosen appropriately].


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