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 *a*‘s are 1 and all the *b*‘s are −1 then this post shows that *D*_{n} = *F*_{n}, the *n*th 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 Poincaré 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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