Generalized Gregory’s series

In this paper the countable family of generalized Gregory’s series (all of which are the power series having the radius of convergence equal to one) is defined. Sums Sn(x),n=1,2,… of these series are found – first in the definite integral form and next in the form of finite linear combination of arcus tangents of some linear forms of x  . It enables to extend the definition of Sn(x)Sn(x) for all x∈C⧹-12n for every n∈Nn∈N. Moreover, for new functions Sn(x)Sn(x), obtained in result of this extension, many fundamental and surprising integral and trigonometric identities are received. Sums of the series of differences of odd harmonic numbers are presented as well.