Compact Convergence of a Power Series (7.46.5)

James Munkres Topology, chapter 7.46, exercise 5:

Consider the sequence of functions $f_n : (-1,1)→ℝ$ defined by $$f_n(x)=\sum_{k=1}^n kx^k$$ (a) Show $(f_n)$ converges in the topology of compact convergence; conclude that the limit function is continuous.
(b) Show $(f_n)$ does not converge uniformly.

Proof: (a) First, note that $f(x)=\sum kx^k$ converges for all $|x| < 1$ by the ratio test. Since each compact subset of $(-1,1)$ is contained in some interval $[-x,x]⊆(-1,1)$, it will suffice to show $f_n$ converges uniformly on $[-x,x]$ for all $x∈(0,1)$. Therefore let $|y| ≤ |x|$ and observe $$|f(y)-f_n(y)| = |\sum^∞_{k=1}ky^k-\sum^n_{k=1}ky^k| = |\sum^∞_{k=n+1} ky^k| ≤$$ $$\sum^∞_{k=n+1} k|x|^k →0$$ as $n→∞$.

(b) Suppose some $n$ such that $|f(x)-f_n(x)| < 1/2$ for all $x∈(-1,1)$. Simply choose $x∈(0,1)$ so that $(n+1)x^{n+1} ≥ 1/2$, and observe $$|f(x)-f_n(x)| = |\sum_{k=n+1}^∞ kx^k| ≥ 1/2~~\square$$