Perhaps you are right that this kind of modification still converges to the natural solution. However there must be ways to solve the infinite equation system that yield different solutions.
What for example if we form the matrix \( E_{|N} \) for which \( (E_{|N}-I)^{-1}(1,0,\dots)^T=\vec{\text{slog}}_{|N} \) not by simply truncating the infinite matrix \( E \) (after stripping the first column) to NxN but by some other methods of selecting the entries of a NxN square matrix.
But I am not sure in which other ways one can choose this matrix, so that the
1. The solution converges for N->oo
2. The resulting solution indeed satisfies the infinite system.
Edit: I just look a very interesting video about solutions of infinite linear equation systems. Very interesting for our case. See here, look the video, not only the slides.
What for example if we form the matrix \( E_{|N} \) for which \( (E_{|N}-I)^{-1}(1,0,\dots)^T=\vec{\text{slog}}_{|N} \) not by simply truncating the infinite matrix \( E \) (after stripping the first column) to NxN but by some other methods of selecting the entries of a NxN square matrix.
But I am not sure in which other ways one can choose this matrix, so that the
1. The solution converges for N->oo
2. The resulting solution indeed satisfies the infinite system.
Edit: I just look a very interesting video about solutions of infinite linear equation systems. Very interesting for our case. See here, look the video, not only the slides.