  In this paper we present a new mathematical conception based on a new method
for ordering the integers. The method relies on the assumption that negative
numbers are beyond infinity, which goes back to Wallis and Euler. We also
present a new axiom system, the model of which is arithmetics. We define
regular method for summation of infinite series which allows us to discover
general and unified approach to summation of divergent series, and determine
the limits of unbounded and oscillating functions. Several properties for
divergent series and explicit formulas for sums of some infinite series are
established. A number of finite and new recurrence formulas for Bernoulli
numbers are obtained. We rederive some known results, but in a simpler and
elementary way, and establish new results by means of techniques of the
theoretical background developed.
