Let's study the convergence (pointwise and uniform convergence) of of the sequence of functions: $$ f_n(x)=n^2 \cdot \log{(1+\frac{x}{n})}, \; x\ge0, \; n \in \Bbb N $$ My attempt:
- I first study: $\lim_{n \rightarrow +\infty}f_n{(x)}$ for pointwise convergence, and i get: $$ f_n{(x)} \rightarrow \begin{cases} 0, \; \text{if} \; \; x=0 \\[2ex] +\infty, \; \text{otherwise}. \end{cases} $$
- In conclusion, the sequences does not converge: nor pointwisely, neither uniformly.
Although I'm pretty sure of that result (if not I have misunderstood the whole theory), it seems weird to me. Any advices?