Question: Consider $\Phi$ to be the space of sequences that have finitely many non-zero terms.
The space is not closed in $\ell\_1$, therefore $\ell\_1/\Phi$ with the quotient topology is not Hausdorff, and so it cannot be metrizable. However, does there exist a metric on $\ell\_1/\Phi$ that gives rise to a non-trivial topology. And even stronger question is, is $\ell\_1/\Phi$ normable?

Answer: 
I would use $$\sum\_{i=1} ^n (i+2)! - i! = (n+2)!+(n+1)! - 2! - 1 !$$
As André Nicolas notes, the essence of the identity is showing that 
$$\sum\_{k=1}^n k \cdot k! = (n+1)! - 1.$$

There's a nice combinatorial proof of this. (See Benjamin and Quinn, *[Proofs that Really Count](http://books.google.com/books?id=n8Xx2oSlz4MC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false)*, Identity 181 on p. 92.) I'll give it in its $\sum\_{k=1}^{n-1} k \cdot k! = n! - 1$ form. 

Both sides count the number of permutations of $1, 2, \ldots, n$ that exclude the identity permutation. 

The right side is straightforward. 

For the left side, how many permutations have $n-k$ as the first number that does not get mapped to itself? There are $k$ choices ($n-k+1, n-k+2, \ldots, n$) for the number that appears in position $n-k$, and then there are $k!$ ways to choose the remaining $k$ numbers to complete the permutation. Adding up over all possible values of $k$ yields the left-hand side.