Sum over k from 2 to Infinity of Zeta of k Minus One

Proof
Sum down each column, then sum across:


 * $\begin{array}{r|cccccccccc}

\paren {\map \zeta k - 1} & &  &  &  &  &  &  \\ \hline

\paren {\map \zeta 2 - 1} & \paren {\dfrac 1 2}^2 & \paren {\dfrac 1 3}^2 & \paren {\dfrac 1 4}^2 & \paren {\dfrac 1 5}^2 & \cdots & \paren {\dfrac 1 n}^2 & \cdots \\

\paren {\map \zeta 3 - 1} & \paren {\dfrac 1 2}^3 & \paren {\dfrac 1 3}^3 & \paren {\dfrac 1 4}^3 & \paren {\dfrac 1 5}^3 & \cdots & \paren {\dfrac 1 n}^3 & \cdots \\

\paren {\map \zeta 4 - 1} & \paren {\dfrac 1 2}^4 & \paren {\dfrac 1 3}^4 & \paren {\dfrac 14}^4 & \paren {\dfrac 1 5}^4 & \cdots & \paren {\dfrac 1 n}^4 & \cdots \\

\paren {\map \zeta 5 - 1} & \paren {\dfrac 1 2}^5 & \paren {\dfrac 1 3}^5 & \paren {\dfrac 1 4}^5 & \paren {\dfrac 1 5}^5 & \cdots & \paren {\dfrac 1 n}^5 & \cdots \\

\cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \hline \sum_{k \mathop = 2}^\infty \paren {\map \zeta k - 1} & \paren {\dfrac 1 1} \paren {\dfrac 1 2} & \paren {\dfrac 1 2} \paren {\dfrac 1 3} & \paren {\dfrac 1 3} \paren {\dfrac 1 4} & \paren {\dfrac 1 4} \paren {\dfrac 1 5} & \cdots & \paren {\dfrac 1 {n - 1} } \paren {\dfrac 1 n} & \cdots \end{array}$

First, summing down the array:

Next, summing across, we note:

Therefore,

Hence: