Uniform convergence of functional sequence $f_{k}=\frac{1}{k^{3}}\ln(1+k^{4}x^{2})$

Question

Is $f_{k}=\frac{1}{k^{3}}\ln(1+k^{4}x^{2})$ uniformly convergent? I know that $$f(x)=\lim_{k\to\infty}f_{k}(x)=\lim_{k\to\infty}\frac{1}{k^{3}}\ln(1+k^{4}x^{2})=\lim_{k\to\infty}\frac{\ln(1+k^{4}x^{2})}{k^{3}}=0, \ \forall x\in\mathbb{R}.$$ So this sequence is convergent when $x\in\mathbb{R}$, but how to check if it is uniformly convergent or not?

Answer 1

For $x_k = e^{k^3}$ with $k \in \mathbb N$ you have

$$1 + k^4x_k^2 > e^{k^3}$$

and therefore $f_k(x_k) > 1$. Hence $\{f_k\}$ doesn't converge uniformly as the pointwise limit is the always vanishing map.