Consider a sequence of functions $f_n : (0,\infty) \rightarrow \Bbb R$ defined by

$$f_n(x)=\frac{n}{n + x + nx^2}$$

Show that $f_n(x)\le f_{n+1}(x)$ for all $n \in \Bbb N$ and $x \in (0,\infty)$. Hence, compute

$$\lim_{n\to \infty}\int_{(0,\infty)}\frac{n}{n+x+nx^2}$$

  • $\begingroup$ Please provide additional context, which ideally explains why the question is relevant to you and the community. Some forms of context include background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc. $\endgroup$ – Sahiba Arora Jun 2 at 15:05

$$f_n(x)=\frac{1}{1 + x^2 + \frac{x}{n}}\lt \frac{1}{1 + x^2 + \frac{x}{n+1}}=f_{n+1}(x)$$ and $$f_n(x) \lt \frac{1}{1+x^2} \text{ for all $n\ge 1, x \gt 0$ }$$ Since $f_n(x)\to f(x) = 1/(1+x^2)$ pointwise and $f$ is integrable, monotone convergence gives $$\lim_{n\to \infty}\int_0^\infty\frac{n}{n+x+nx^2} = \int_0^\infty\frac{1}{1+x^2} = \frac{\pi}{2}$$

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