Definition:Infinite Hilbert Matrix

Definition
The infinite Hilbert matrix is the infinite matrix whose elements are defined as:
 * $a_{ij} = \dfrac 1 {i + j - 1}$

In full, it appears as:
 * $\begin{bmatrix}

1 & \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \ldots \\ \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \ldots \\ \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \ldots \\ \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \ldots \\ \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} & \ldots \\ \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} & \tfrac 1 {11} & \ldots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix}$