real analysis - Lebesgue integrable function?

If $\displaystyle f(x)=\frac{1}{x^p}$ $(0 < x \leq 1)$ then $f \in L[0,1]$ if $p<1$ and

$$\int_{0}^1 f= \frac{1}{1+p}$$

I know that non negative measurable function f is Lebesgue integrable on [a,b] if

$$\int_{a}^b f=\lim_{n \to \infty} \int_{a}^b f^n$$
If this limit is finite then function is Lebesgue integrable.

but how can i find $f^n$ for this function? please help me.Thanks in advance.