Does anyone know where to find a proof that these nested symmetric matrices are non singular? Trivial for the first two, written below. But then?
$$A_1 = \begin{pmatrix} 1 & 1/2 \\ 1/2 & 1/3\\ \end{pmatrix}$$
$$ A_2 = \begin{pmatrix} 1 & 1/2 & 1/3 \\ 1/2 & 1/3 & 1/ 4 \\ 1/3 & 1/4 & 1/5\\ \end{pmatrix}$$
..... $$ A_P= \begin{pmatrix} 1 & \frac{ 1}{2} & \frac{ 1}{3} & \cdots & \frac{ 1}{ P+1} \\ \frac{ 1}{ 2} & \frac{ 1^{\vphantom{P^-}}}{ 3} & \frac{ 1}{ 4} & \cdots& \frac{ 1}{ P+2} \\ \frac{ 1}{ 3} & \frac{ 1^{\vphantom{P^-}}}{ 4} & \frac{ 1}{ 5} & \cdots& \frac{ 1}{ P+2} \\ \vdots\\ \frac{ 1}{ P+1} & \frac{ 1}{ P+2} & \frac{ 1}{ P+3} & \cdots &\frac{ 1}{ 2P+1}\end{pmatrix}$$
