Matrix question for Gottfried

[Gottfried]

While we are here, you mentioned geometrical series of matrices. I work with geometrical series and want my work to handle matrices. Here is my totally stupid problem, I couldn't quickly find a closed for for the geometrical series of matrices while the geometrical series of reals is a high school problem.

Hi Daniel -

 I've got involved into geometric series of matrices when I experimented with divergent summation. In general one can use that geometric series (= series -of-powers-of-matrix) by applying diagonalization. Having \( B =   M \cdot D \cdot W \) (where \( W = M^{-1} \) for notational comfort) and \( D \) is diagonal, then the geometric series of \( B \) is \( M \cdot (D^0+D^{1}+D^{2}+ ... ) \cdot W \) and the geometric series of the diagonalmatrix \( D \) is represented by the geometric series in the elements of the diagonal matrix \( D \) (one might write for the alternating geometric series \( (I + B)^{-1} = M \cdot (I+D)^{-1} \cdot W \) and the alternating geometric series of \(D\) can be expressed by the alternating geometric series of the elements on its diagonal.  

With Carleman-matrices it is in principle impossible to define this geometric series, because (at least) one eigenvalue is \( 1 \) by construction, and we had one entry \( 1 / 0 \) in one entry. Thus I mostly worked with the alternating geometric series in our (or my other) contexts.   
There is one little, but I think significant, exception, and this is the geometric series of the pascalmatrix \( P \) which performs \( V(x) \to V(x+1) \) . I found a workaround for the geometric series via diagonalization (we cannot define a proper diagonalization on \( P \) , but I found somehow an improper, but useful, one) and this led to the rediscovery of the Faulhaber-matrix, (or matrix of integrals of the Bernoulli-polynomials), with which I can implement Hurwitz zeta. That has one aspect which made me curious whether perhaps we overlook something relevant in our common calculations. (See index-page https://go.helms-net.de/math/index.htm and entries https://go.helms-net.de/math/binomial_ne...Powers.pdf and https://go.helms-net.de/math/binomial_ne...Laurin.pdf)

So far at the moment, it's late here, and soon I'll invite sandman ;-) If you need more input, I come back to this tomorrow.

Gottfrie