How to use the Lebesgue Dominated Convergence theorem,
where $f_n(x) = \displaystyle\frac{nx^n}{1+x}$ and domain is [0,1].
I want to find $\lim_{n \to \infty}\int^1_0 f_n(x) dx$, I will use the Lebesgue Dominated Convergence theorem.
May be I think I must use integrate by part for finding solution…
Best Answer
Use integration by parts to show that (integrate $nx^n$ and diffrentiate $1/(1+x)$):$$\int^1_0 nx^n/(1+x)= n/2(n+1)+\int_0^1 (n/(n+1))x^{n+1}/(1+x)^2dx$$ Now use Lebesgue's dominated convergence to find : $$\lim_{n \to \infty}\int_0^1 (n/(n+1))x^{n+1}/(1+x)^2dx$$ The sequence of functions $(n/(n+1))x^{n+1}/(1+x)^2$ is dominated by the integrable function $1/(1+x)^2$. Since $\lim_{n \to \infty}(n/(n+1))x^{n+1}/(1+x)^2=g(x)$ (where $g(x)=0$ if $x<1$, $g(1)=1/4$), thus: $$\lim_{n \to \infty}\int_0^1 (n/(n+1))x^{n+1}/(1+x)^2dx=\int_0^1g(x)dx=0$$
Therefore: $$\lim_{n \to \infty}\int^1_0 nx^n/(1+x)=\lim_{n \to \infty}n/2(n+1)=1/2$$