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[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.

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First, note that independently of $k$, the sequence $\{f_n\}$ converges pointwise to $0$.

For the first question, if $f_n\to 0$ uniformly in the real line, then $f_n(n)$ has to converge to $0$, hence $\frac{n^k}{n^2+n}\to 0$ so $k< 2$. Conversely, if $k= 1$, then for all $x$, $$|f_n(x)|=\frac{|x|\sqrt n\frac 1{\sqrt n}}{x^2+n}\leqslant \frac 1{2\sqrt n}.$$

For the second question, we have, if $|x|\leqslant A$, that
$$|f_n(x)|\leqslant \frac{A^k}n,$$ so for all $k$, the convergence is uniform on bounded sets.

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