The most common way to prove uniform convergence of a function series is to dominate its terms by a convergent numerical series. The focus keyword Weierstrass M-test names this method. It is simple, reliable, and widely used for trigonometric, power-type, and step-function series. :::theorem[Weierstrass M-Test] Let $f_n$ be real-valued functions on $X$. Suppose $|f_n(x)|\leq M_n$ for all $x\in X$ and all $n$, where $M_n\geq0$. If $\sum M_n$ converges, then $\sum f_n$ converges uniformly on $X$. ::: :::proof Given that $|f_n(x)|\leq M_n$ and $\sum M_n$ converges. To prove uniform convergence. Let $\epsilon>0$. Since $\sum M_n$ converges, choose $N$ such that $\sum_{k=n+1}^mM_k<\epsilon$ for all $m>n\geq N$. Then for every $x\in X$, $|\sum_{k=n+1}^mf_k(x)|\leq\sum_{k=n+1}^m|f_k(x)|\leq\sum_{k=n+1}^mM_k<\epsilon$. Thus the partial sums are uniformly Cauchy, so the series converges uniformly. $\square$ ::: :::example[Trigonometric Series] For $p>1$, the series $\sum_{n=1}^{\infty}\sin(nx)/n^p$ converges uniformly because $|\sin(nx)/n^p|\leq1/n^p$ and $\sum1/n^p$ converges. ::: :::solved-problem Prove that $\sum_{k=1}^{\infty}((x/(1+x))\sin x)^k$ is uniformly convergent on $[0,1]$. ::: :::solution Let $h(x)=(x/(1+x))\sin x$. For $x\in[0,1]$, $0\leq x/(1+x)\leq1/2$ and $0\leq\sin x\leq\sin1$. Hence $|h(x)|\leq(\sin1)/2<1$. Therefore $|h(x)^k|\leq((\sin1)/2)^k$, and the corresponding geometric series converges. By the Weierstrass M-test, the function series converges uniformly. ::: :::solved-problem Let $I(x)=0$ for $x\leq0$ and $I(x)=1$ for $x>0$. Let $x_n$ be distinct points in $(a,b)$ and suppose $\sum|c_n|$ converges. Prove uniform convergence of $\sum c_nI(x-x_n)$ on $[a,b]$ and continuity away from the $x_n$. ::: :::solution For every $x\in[a,b]$, $|c_nI(x-x_n)|\leq|c_n|$. Since $\sum|c_n|$ converges, the M-test gives uniform convergence. At a point different from all $x_n$, each step function is continuous locally, and the uniform convergence of the series preserves continuity at that point. ::: :::remark The M-test proves absolute uniform convergence. It is sufficient but not necessary. ::: :::exercise 1. Prove uniform convergence of $\sum x^n/n^2$ on $[0,1]$. 2. Why is $p>1$ needed for $\sum\sin(nx)/n^p$ by this test? 3. Give a majorant for $\sum\cos(nx)/3^n$. ::: :::answer 1. Use $|x^n/n^2|\leq1/n^2$. 2. Because $\sum1/n^p$ converges only for $p>1$. 3. Use $M_n=1/3^n$. :::