[Berkeley PhD Qualifying Exam question, 1978]
Let $k \ge 0$ be an integer and define a sequence of maps
$f_n : R \to R, \ \ \ \ f_n(x) =\frac{x^k}{x^2+n}, \ \ \ \ n=1,2,3,\dots $
a) For which values of $k$ does the sequence converge uniformly on $R$?
b) For which values of $k$ does the sequence converge uniformly on every bounded subset of $R$?
Attempt at Solution:
For part (a), the sequence converges uniformly for $0 \le k \le 2$. For part (b), I see that $n \to \infty$ can now dominate powers of $x$ on a bounded subset of $R$, so I think that the range of $k$ for uniform convergence should be larger.