A function series is studied through its partial sums. The focus keyword uniformly convergent series refers to the uniform convergence of these partial sums. This condition is strong enough to preserve continuity of the sum when each term is continuous. It is a central method in approximation theory and analysis. :::definition[Convergence of a Series of Functions] Let $f_1,f_2,\ldots$ be real-valued functions on $X$. The series $\sum_{n=1}^{\infty}f_n$ converges on $X$ to $f$ if the partial sums $S_n=f_1+\cdots+f_n$ converge pointwise to $f$. ::: :::example[Geometric Function Series] For $-1<x<1$, the series $\sum_{n=1}^{\infty}x^n$ converges pointwise to $x/(1-x)$. ::: :::definition[Uniform Convergence of a Function Series] The series $\sum_{n=1}^{\infty}f_n$ converges uniformly on $X$ to $f$ if the partial sums $S_n=f_1+\cdots+f_n$ converge uniformly to $f$ on $X$. ::: :::corollary Let $f_1,f_2,\ldots$ be continuous real-valued functions on a metric space $X$. If $\sum f_n$ converges uniformly to $f$ on $X$, then $f$ is continuous on $X$. ::: :::proof Given that the series converges uniformly and each term is continuous. To prove continuity of the sum. The partial sums $S_n=f_1+\cdots+f_n$ are continuous because they are finite sums of continuous functions. Since $S_n\to f$ uniformly, the uniform limit theorem gives that $f$ is continuous. $\square$ ::: :::example[Non-Uniform Series with Discontinuous Sum] On $[0,1]$, the series $\sum_{n=0}^{\infty}x(1-x)^n$ has sum $0$ at $x=0$ and $1$ for $0<x\leq1$. The sum is discontinuous, so the convergence cannot be uniform because each term is continuous. ::: :::solved-problem Show that $\sum_{n=1}^{\infty}x^n/2^n$ converges uniformly on $[0,1]$ and find its sum. ::: :::solution For $x\in[0,1]$, $0\leq x^n/2^n\leq1/2^n$, and $\sum 2^{-n}$ converges. Hence the convergence is uniform by the Weierstrass M-test. The sum is $\sum (x/2)^n=(x/2)/(1-x/2)=x/(2-x)$. ::: :::remark Uniform convergence of a series is a statement about the tails of partial sums, not merely about the individual terms. ::: :::exercise 1. Define partial sums. 2. Why does a uniformly convergent series of continuous functions have continuous sum? 3. Find the pointwise sum of $\sum_{n=0}^{\infty}x^n$ on $(-1,1)$. ::: :::answer 1. $S_n=f_1+\cdots+f_n$. 2. Partial sums are continuous and converge uniformly. 3. The sum is $1/(1-x)$. ::: :::faq[Frequently Asked Questions] Q: Is uniform convergence of a series the same as uniform convergence of terms? A: No. It means uniform convergence of partial sums. Q: Why does continuity pass to the sum? A: The sum is the uniform limit of continuous partial sums. :::