Irrationality Proofs: From e to Zeta(n>=2)

We develop definitions and a theory for convergent series that have terms of the form 1/a_j where a_j is an integer greater than one and the series convergence point is less than one. These series have terms with denominators that can be used as number bases. The series for e-2 and z_n=\zeta(n)-1 are of this type. Further, both series yield number bases that can represent all possible rational convergence points as single digits. As partials for these series are rational numbers, all partials can be given as single decimals using some a_j as a base. In the case of e-2, the last term of a partial yields such a base and partials form systems of nesting inequalities yielding a proof of the irrationality of e-2. In the case of z_n, using the z_2 case we determine that such systems of nesting inequalities are not formed, but we discover partials require bases greater than the denominator of their last term. We prove this for the general z_n and using it we give a proof of the irrationality of all z_n.