I am studying about power series on topic Series of Functions and i am trying to solve some exercise. I see exapmle 1 to solve my problem. Here the question :
Express the Lebesgue integral $$F(x)= \int_{(0,1)} \frac {1}{1-x^3}dx$$ as a power series.
My attempt :
From the function $f(x)=\frac1{1-x^3}$, I got the power series representation $f(x)= \sum_{n=0}^\infty (x^3)^n$.
Then what is the next step ?
$f_m = \sum_{n=0}^{m} x^{3n} $ is a sequence of monotone measurable functions on $[0,1]$ which converge to $\frac{1}{1-x^3}$ The monotone convergence theorem allows for
$$\int_{0}^{1} \sum_{n=0}^{\infty} x^{3n}dx = \int_{0}^{1} \lim_{m \to \infty}\sum_{n=0}^{m} x^{3n}dx = \lim_{m \to \infty}\int_{0}^{1} \sum_{n=0}^{m} x^{3n} dx = \sum_{n=0}^{\infty}\frac{1}{3n+1} $$