The article is devoted to the expansion of iterated Stratonovich stochastic integrals of second multiplicity into the double series of products of standard Gaussian random variables. The proof of expansion is based on the application of double Fourier-Legendre series and double trigonometric Fourier series summarized by Pringsheim method. The results of the article are generalized to the case of an arbitrary complete orthonormal system of functions in the space $L_2([t, T])$ and $ψ_1(τ), ψ_2(τ)\in L_2([t, T]).$ The considered expansion can be applied to the numerical integration of Ito stochastic differential equations. Some recent results on the expansion of iterated Stratonovich stochastic integrals of multiplicities 3 to 6 are given.