While learning spectral sequence, I have trouble realizing where the terms $E^k$ get stationary $E^\infty$. For bounded chain complexes, it happens since the filtration is finite. But, for example, the Basic pathspace fibration in this wiki page states that $E^2=E^\infty$ and atually use the $d_{n+1}$ in page $E^{n+1}$. I have not came across the proof of Larey-Serre sequence though. I wonder if it's related to a filtration.
Would you please explain why there is $E^2=E^\infty$ in this example?
The Serre spectral sequence for $\Omega S^n \to PS^n \to S^n$ does not degenerate, i.e., $E^2 \neq E^\infty$. If it did, the projection $PS^n \to S^n$ would induce a surjective in homology which is absurd since $PS^n$ is contractible but $S^n$ has nontrivial homology in degree $n$.
Indeed, as Wikipedia explains, there is a $d^n$-differential that kills the class in $H_n(S^n)$. After that, there are no more possible nontrivial differentials, so the spectral sequence collapses and $E^{n+1} = E^\infty$.