Beta Function: The beta function B(m,n) = $\int_0^1$ xm-1(1-x)n-1 dx converges when m,n >0. Gamma Function: The gamma function Γ(n) = $\int_0^\infty$ e-xxn-1dx converges when n >0. Properties of Beta and Gamma Functions: The list of the properties of beta and gamma functions is given below: P1: B(m,n) = B(n,m) P2: B(m,n) = 2$\int_0^{\pi/2}$ sin2m-1θ … Read more
In this post, we will learn about Abel’s theorem (also known as Abel-Pringsheim’s Theorem) along with a few questions and answers. Statement of Abel’s Theorem: If {an} is a monotone decreasing sequence of positive terms, then limn→∞ nan= 0 is a necessary condition for the convergence of the series ∑an. Corollary: If {an} is a … Read more
In this post, we will learn about Cauch’s criterion for the convergence of a series. The statement of Cauch’s criterion for series along with some applications will be provided here. Statement of Cauch’s criterion for series Statement: The series $\sum_{n=1}^\infty a_n$ converges if and only if given any ε>0, there exists a positive integer N … Read more
In this post, we will learn about Rolle’s theorem along with examples. We will also discuss some of the related questions of Rolle’s theorem with answers. Rolle’s Theorem Statement Let f(x) be a real-valued function defined on [a, b]. If then there exists at least one element c in (a, b) such that $f'(c)=0$. Examples … Read more