The Weierstrass M-Test for Uniform Convergence of Series of Functions

# The Weierstrass M-Test for Uniform Convergence of Series of Functions

Recall from the Pointwise Convergent and Uniformly Convergent of Series of Functions page that if $(f_n(x))_{n=1}^{\infty}$ is a sequence of real-valued functions with common domain $X$, then we say that the corresponding series of functions $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ is uniformly convergent if the sequence of partial sums $(s_n(x))_{n=1}^{\infty}$ is a uniformly convergent sequence.

We will now look at a very nice and relatively simply test to determine uniform convergence of a series of real-valued functions called the Weierstrass M-test.

 Theorem 1: Let $(f_n(x))_{n=1}^{\infty}$ be a sequence of real-valued functions with common domain $X$, and let $(M_n)_{n=1}^{\infty}$ be a sequence of nonnegative real numbers such that $\mid f_n(x) \mid \leq M_n$ for each $n \in \mathbb{N}$ and for all $x \in X$. If $\displaystyle{\sum_{n=1}^{\infty} M_n}$ converges then $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ uniformly converges on $X$.
• Proof: Suppose that there exists a sequence of nonnegative real numbers $(M_n)_{n=1}^{\infty}$ such that for all $n \in \mathbb{N}$ and for all $x \in X$ we have that:
(1)
\begin{align} \quad \mid f_n(x) \mid \leq M_n \end{align}
• Furthermore, suppose that $\displaystyle{\sum_{n=1}^{\infty} M_n}$ converges to some $M \in \mathbb{R}$, $M \geq 0$. Then we have that for all $x \in X$:
(2)
\begin{align} \quad \biggr \lvert \sum_{n=1}^{\infty} f_n(x) \biggr \rvert \leq \sum_{n=1}^{\infty} \mid f_n(x) \mid \leq \sum_{n=1}^{\infty} M_n = M \end{align}
• So the $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ converges for each $x \in X$ by the comparison test. $\blacksquare$