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}

\displaystyle \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 {1} {4}}^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,