I have some trouble to understand the difference between the p-completion and p-local space. if $X$ a simply connected spaces such that all higher homotopy groups are finitely generated groups then the p-local approximation $X\rightarrow L_{p}X$ coincides with the $p$-completion $X_{p}^{\wedge}$. Is it correct? Now if $X$ is not simply connected such that the fundamental group is abelian do we have that $ L_{p}X \simeq X_{p}^{\wedge}$. I think it is true if $X$ is a circle, more precisely $S^{1\wedge}_{p}\sim L_{p}S^{1}\sim K(\mathbf{Z}_{p}^{\wedge},1)$. Am I wrong ?