Determinant of an infinite matrix

What does the infinite determinant

mean and when does it converge?

The determinant D above is the limit of the determinants D_n defined by

If all the as are 1 and all the bs are -1 then this post shows that D_n = F_n, the nth Fibonacci number. The Fibonacci numbers obviously don't converge, so in this case the determinant of the infinite matrix does not converge.

In 1895, Helge von Koch said in a letter to Poincare that the infinite determinant is absolutely convergent if and only if the sum

is absolutely convergent. A proof is given in [1].

The proof shows that the D_n are bounded by

and so the infinite determinant converges if the corresponding infinite product converges. And a theorem on infinite products says

converges absolute if the sum in Koch's theorem converges. In fact,

and so we have an upper bound on the infinite determinant.

Related post: Triadiagonal systems, determinants, and cubic splines

[1] A. A. Shaw. H. von Koch's First Lemma and Its Generalization. The American Mathematical Monthly, April 1931, Vol. 38, No. 4, pp. 188-194

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