# Find the inverse of this infinite matrix

Can anyone find the inverse of the following infinite matrix?

$M=\begin{bmatrix} ^1 /_2 & ^1/_3 & ^1/_4 & \cdots\\ ^1 /_3 & ^1/_4 & ^1/_5 &\cdots\\ ^1 /_4 & ^1/_5 & ^1/_6 & \cdots\\ \vdots & \vdots & \vdots & \ddots \end{bmatrix}, \qquad M^{-1}= \ ?$

• A (finite dimensional) analogue of this is the Hilbert Matrix. There are clearly some differences, but it might be useful. It also appears to be a Hankel Matrix – Mark Nov 13 '16 at 20:57
• This also appears to be a Cauchy Matrix, which seems to have an inversion formula (For finite dimensional matrices). – Mark Nov 13 '16 at 21:01
• @Mark Is there any reason to expect that these inversion formulas should generalise to an infinite matrix? – user1892304 Nov 13 '16 at 21:14
• No clue, I had just remembered seeing the Hilbert Matrix before, and that this looks similar. I'm unfamiliar with infinite dimensional matrices though. – Mark Nov 13 '16 at 21:31