For non-negative integers $k\geq 1$ define $$ f_k(x) = \frac{x^k}{(1+x)^2},~x\geq 0. $$ Which of the following statements are true?
- For each $k$, $f_k$ is a function of bounded variation on compact intervals
- For every $k$, $\int_0^1 f_k(x)~\mathrm{d} x < \infty $
- $\lim_{k\to \infty } \int_0^1 f_k(x) \mathrm{d} x$ exists
- The sequence of functions $f_k$ converge uniformly on $[0,1]$ as $k\to \infty $
Note that in (4), we have $$f(x) = \begin{cases}0,~\text{ if }x<1 \\ 1/4,~\text{ if }x=1,\end{cases}$$ which is not continuous, and hence the convergence is not uniform.
For option (2), each of the functions is continuous and hence bounded on $[0,1]$ and therefore, the integral $\int_0^1 f_k(x)~\mathrm{d}x$ will be finite.
I got stuck with options (1) and (3). Kindly help me!