Some properties of Chebyshev's series are discussed. These series are used as a basis for constructing numerical analytical methods of solving Cauchy problems for systems of ordinary differential equations. Particular attention is given to the calculation of Chebyshev's coefficients with the aid of numerical integration. A Markov quadrature formula with two fixed nodes and the weight function that corresponds to the orthogonal system of Chebyshev's shifted polynomials of the first kind is derived. Some properties of partial sums of Chebyshev's series with the coefficients obtained by Markov's formula are described.

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