Answer to Question #170533 in Real Analysis for Prathibha Rose

Question

State and prove the weierstrass M -test for the uniform convergence of a series of functions

Answer 1

Statement: A series $\sum^{\infin}_1U_n(x)$ of functions will converge uniformly on I. If there exist a convergent series $\sum^{\infin}_1M_n$ of positive constant such that $|U_n(x)|\eqslantless M_n \forall n \in N and \space \forall \in I$

Proof: $\sum M_n$ is convergent

By Cauchy's principle for given $\in >0 \& m \in N$ such that $M_{n+1}+M_{n+2}+.....+M_{n+p} < \in ..........1$

$\forall n \eqslantgtr m; p=1,2,3$

Our hypothesis $|U_n(x)|\eqslantless M_n \forall n \in N and \space \forall \in I$

Now $|U_{n+1}(x)+U_{n+2}(x)+...+U_{n+p}(x)| \eqslantless |U_{n+1}(x)|+|U_{n+2}(x)|+...+|U_{n+p}(x)| \eqslantless M_{n+1}+M_{n+2}....+M_{n+p} < \in \forall n \eqslantgtr m , p=1,2...$

$|U_{n+1}(x)+U_{n+2}(x)+...+U_{n+p}(x)| < \in \forall n < m , p=1,2...$

Therefore $U_n(x)$ convergent uniformly and absolutely on I