sum(e - eta^^k): convergence or divergence?

[bo198214]

using \( \eta = e^{1/e} = 1.44466786... \)

the sum:

\( S = \sum_{k=0}^{\infty} ( e- \eta\^ \^^k ) \)

Clearly the sequence of terms tends to zero because e is the fixpoint of iteration and in a first guess I thought that also the series converges. But the convergence of the sequence is slow and one needs a lot of terms to see a promising trend.

Perhaps you can post the open problem in the open problems survery.
I dont think it is too difficult, however didnt find a proper proof yet.