What is the Homology Spectral Sequence for a Chain Complex with Filtration?

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Here is this week's POTW:

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Let $\mathcal{C} = \{C_n\}_{n\ge 0}$ be a chain complex. Compute the homology spectral sequence associated with the filtration $F_m := \sum_{n = 0}^m C_n$, $m\ge 0$.
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No one answered this week's problem. You can read my solution below.

Spoiler

I'll let $d$ represent the differential of the chain complex. Since $E_n^0 \approx F_n/F_{n-1} \approx C_n$ and $d(F_n) \subseteq F_{n-1}$, then the map $d^0_n : F_n/F_{n-1} \to F_n/F_{n-1}$ is zero, $(E^0,E_n^0,d^0)$ is identified with $(\mathcal{C},C_n,0)$, and $E_n^1 \approx E_n^0 \approx C_n$. The map $d_n^1 : F_n/F_{n-1} \to F_{n-1}/F_{n-2}$ reduces to $d_n : C_n \to C_{n-1}$, and so $(E^1,E_n^1,d^1)$ is identified with $(\mathcal{C},C_n,d_n)$. We have $E_n^2 \approx H_n(\mathcal{C})$ and $d^2_n$ is zero, so $(E^2, E_n^2, d^2)$ is identified with $(H_*(\mathcal{C}), H_n(C), 0)$. All higher order $d^r$ are zero, $E_n^r \approx H_n(\mathcal{C})$ for $2 \le r < \infty$. Finally, $E_n^\infty \approx \sum_{m = 0}^n H_m(\mathcal{C})/\sum_{m = 0}^{n-1} H_m(\mathcal{C}) \approx H_n(\mathcal{C})$.