Let <$E f~symbol <174>~C~inf~[-1, 1]> and <$E symbol $~ rho~ symbol <174>~[1, 2)> such that <$E symbol <34> k~=~0, 1, 2, .... ||f sup (k) || sub C[-1,1]~symbol Г~ c(f) rho sup k 2 {k(k+1) over 2}>. Then it expands in the generalized Taylor series, which was introduced by V. A. Rvachov in 1982. In this paper it is shown that if the restrictions <$E ||f sup (n) ||~=~omicron (2 {n(n+1) over 2})>, <$E n~symbol О~1> are imposed on the sum of this series, and stronger restrictions <$E |f sup (n)~(x sub n,k ) |~symbol Г~CA(n),~A(n+1) over A(n)~symbol Г~2 sup {n+1/2}> hold for its coefficients, then these stronger restrictions will hold for the sum of the series too. As a consequence the conditions of belonging to Gevrey class and of real analyticity for the above-mentioned functions are obtained.

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