Consider the sequence of functions $\{f_n \}$ on $\Bbb R$ defined by $$f_n(x) = n \log \left (1 + \frac {x^2} {n} \right ),\ \ x \in \Bbb R$$ for all $n \geq 1.$ Show that the sequence of functions $\{f_n \}$ is not uniformly convergent on $\Bbb R.$
My attempt $:$ First I observe that the sequence of functions $\{f_n \}$ converges pointwise to the everywhere continuous function $f$ defined by $$f(x) = x^2,\ \ x \in \Bbb R.$$
If the sequence of functions $\{f_n \}$ converges uniformly to the continuous limit function $f$ on whole of $\Bbb R$ then in particular the sequence of functions $\{f_n \}$ converges uniformly to the continuous limit function $f$ on $[0,1].$
But then we have $$\lim\limits_{n \rightarrow \infty} \int_{0}^{1} f_n(x)\ dx = \int_{0}^{1} f(x)\ dx.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$
Now what I found is that $$\int_{0}^{1} f_n(x)\ dx = n \log \left (1 + \frac 1 n \right ) – 2n + 2n^{\frac 3 2} \arctan \left (\frac {1} {\sqrt n} \right ).$$ So we have $$\lim\limits_{n \rightarrow \infty} \int_{0}^{1} f_n(x)\ dx = – \infty.$$ where as $$\int_{0}^{1} f(x)\ dx = \frac 1 3.$$ which is a contradiction to $(1).$ Hence our assumption is false. So the sequence of functions $\{f_n \}$ is not uniformly convergent on $\Bbb R,$ as required.
EDIT $:$ I found the wrong limit. By L'Hospital the limit $$\lim\limits_{n \rightarrow \infty} n \left ( \log \left (1 + \frac 1 n \right ) – 2 \right ) = 1.$$
Now what will be $$\lim\limits_{n \rightarrow \infty} 2n^{\frac 3 2} \arctan \left (\frac {1} {\sqrt n} \right )\ ?$$
By L'Hospital I got infinity. Isn't it the case? But then the overall limit becomes $+\infty.$
Best Answer
Here's an approach: note that $$ f_{n}(x) - f(x) = n \log \left (1 + \frac {x^2} {n} \right ) - \log(e^{x^2}) = \log \left( \frac{(1 + x^2/n)^n}{e^{x^2}} \right). $$ Now, we note that for any fixed $n$, we have $$ \begin{align} \lim_{x \to \infty} \frac{(1 + x^2/n)^n}{e^{x^2}} &= \left(\lim_{x \to \infty} \frac{1 + x^2/n}{e^{x^2/n}}\right)^n = 0 \end{align} $$ (since the numerator is a polynomial) from which we may conclude that (again, for any fixed $n$) the set $\{|f_n(x) - f(x)|: x \in \Bbb R\}$ is unbounded. Thus, if we fix (for instance) $\epsilon = 1$, we can see that for any $n$ there exists an $x$ such that $|f_n(x) - f(x)| \geq \epsilon$, which is to say that $\|f_n - f\|_{\sup} \geq \epsilon$.
So, the sequence is not uniformly convergent.
Regarding your limit: with the substitution $m = \sqrt{1/n}$, we have $$ \begin{align} \lim_{m \to 0^+} 2m^{-3} \arctan \left (m \right ) &= 2\lim_{m \to 0^+}\frac{\arctan(m)}{m^3} = 2\lim_{m \to 0^+}\frac{\frac{1}{m^2 + 1}}{3m^2} = \frac 23 \lim_{m \to 0^+} \frac{1}{m^2(m^2 + 1)} = +\infty \end{align} $$