$ \left( f_k:x \mapsto \chi_k(x) \left(1-\frac{x}{k}\right)^k e^{x/2} \right)_k$ is increasing; [monotone convergence theorem]

Question

Consider the sequence of functions

$$k\mapsto \left( f_k:x \mapsto \chi_k(x) \left(1-\frac{x}{k}\right)^k e^{x/2} \right)$$

where $\chi_k$ is the characteristic function of the interval $[0,k]$ and each $f_k$ is defined over $[0,+\infty).$

Is this sequence increasing?

My reasoning

Let $k<k+1.$ For $x>k+1,$ we have $f_k(x)=f_{k+1}(x)=0;$ for $0\leq x\leq k+1,$ we have $$\left(1-\frac{x}{k+1}\right)^{k+1}\geq \left(1-\frac{x}{k}\right)^k$$ and so $f_{k}(x)\leq f_{k+1}(x).$

Why I am asking the question

I'm reading some notes where the limit $\lim_{k\to +\infty} \int_{[0,k]} \left(1-\frac{x}{k}\right)^k e^{x/2} \ dx$ is evaluated using dominated convergence theorem, while I would have used monotone convergence.

Is there some reason why we can't use monotone convergence theorem as I would have in this case, that I am not seeing?

Answer 1

actually for $x=2.5$ and $k=2$ we have$$\left(1-\frac{x}{k+1}\right)^{k+1} \approx 0.00462963 < 0.0625 \approx \left(1-\frac{x}{k}\right)^{k}$$so your reasoning is incorrect there. I think you actually need dominated convergence.