The hypergeometric Ordinary Differential Equation (ODE) has wide application in the Mechanical Engineering field. The solution to the hypergeometric ODE is called a hypergeometric function, or also a hypergeometric series. Not all of the hypergeometric series converges to a simple defined algebraic function. It is a well-known fact that the hypergeometric series F(1, c, c; x) will converge to the Maclaurin series 1/(1-x). This article investigates the hypergeometric function F(k, c, c; x). It will be proven that the function F(k, c, c; x) will always converge to certain values. We can represent these values on a line of the form A + mx, whose coefficients A and m are functions of x.
Keywords: hypergeometric ODE; hypergeometric function; hypergeometric series.