Chebyshev polynomials, their remarkable properties and connection with Catalan numbers

The main result of the article says that the formal power series equal to the ratio of two neighboring Chebyshev polynomials, after some renormalization, approximates the generating function of the Catalan numbers. We present a proof of this result using continued fractions. Also, for completeness of the presentation, we recall the basic properties of Chebyshev polynomials and formal power series that we use.