Integral convergence

Question

Please how can I show that the $\lim_{n \rightarrow \infty}\int_{\Re^+}f_n d\mu$ converges and determine its limit in the following cases of $f_n: \Re^+ \rightarrow \Re$ (a)$f_n(x) = sin(nx) \chi_{[0,n]}(x)$. (b) $f_n(x)= \dfrac{ne^{-nx}}{\sqrt{1+n^{2}x^{2}}}$. (c)$f_n(x)= \dfrac{ne^{-x}}{\sqrt{1+n^{2}x^{2}}}$.

Answer 1

(a) The integral $\displaystyle \int_0^n \sin nx \, dx$ can be evaluated exactly. Its limit is zero.

(b) Use the elementary substitution $u = nx$, $du = n\, dx$: $$\int_0^\infty \frac{ne^{-nx}}{\sqrt{1 + n^2 x^2}} dx = \int_0^\infty \frac{e^{-u}}{\sqrt{1 + u^2}} du.$$ This integral is finite, so the sequence is constant.

(c) This sequence diverges. You can try direct estimates or use Fatou's lemma: $$\infty = \int_0^\infty \frac{e^{-x}}{x} dx = \int_0^\infty \lim_{n \to \infty} \frac{ne^{-x}}{\sqrt{1 + n^2 x^2}} dx \le \liminf_{n \to \infty} \int_0^\infty \frac{ne^{-x}}{\sqrt{1 + n^2 x^2}} dx.$$