The Gamma function, denoted by the symbol Γ, is a mathematical function that extends the concept of factorial to real and complex numbers. It is defined as the integral of the function e^-x over the interval [0, ∞).
The Gamma function is used to evaluate integrals that cannot be expressed in terms of elementary functions. It is particularly useful when dealing with problems in statistics, probability theory, and number theory.
An integral converges when the value of the integral exists and is finite. In other words, the area under the curve of the function being integrated is well-defined and not infinite.
The Gamma function satisfies the necessary conditions for convergence, such as being continuous and having a positive value over the interval of integration. This allows for the use of various convergence tests, such as the Comparison Test and the Limit Comparison Test.
While the Gamma function is a powerful tool for evaluating integrals, it does have some limitations. For example, it cannot be used for integrals that involve oscillatory functions or functions that do not approach zero as x approaches infinity. In such cases, alternative methods must be used to determine convergence.