\begin{align*} \sum_{k=2}^{\infty}\frac{\zeta\left(k\right)-1}{k} & =\sum_{k=2}^{\infty}\frac{\zeta\left(k\right)\Gamma\left(k\right)-\Gamma\left(k\right)}{k!}\\ & =\sum_{k=2}^{\infty}\frac{1}{k!}\left(\int_{0}^{\infty}\frac{x^{k-1}}{e^{x}-1}dx-\int_{0}^{\infty}x^{k-1}e^{-x}dx\right)\\ & =\sum_{k=2}^{\infty}\frac{1}{k!}\int_{0}^{\infty}x^{k-1}\left(\frac{1}{e^{x}-1}-e^{-x}\right)dx\\ & =\int_{0}^{\infty}\sum_{k=2}^{\infty}\frac{1}{k!}x^{k-1}\left(\frac{1}{e^{x}-1}-e^{-x}\right)dx\cmt{\text{総和と積分の順序変更}}\\ & =\int_{0}^{\infty}\frac{1}{x}\sum_{k=2}^{\infty}\frac{1}{k!}x^{k}\left(\frac{1}{e^{x}-1}-e^{-x}\right)dx\\ & =\int_{0}^{\infty}\frac{1}{x}\left(\sum_{k=0}^{\infty}\frac{1}{k!}x^{k}-\left(1+x\right)\right)\left(\frac{1}{e^{x}-1}-e^{-x}\right)dx\\ & =\int_{0}^{\infty}\frac{1}{x}\left(e^{x}-1-x\right)\left(\frac{1}{e^{x}-1}-e^{-x}\right)dx\\ & =\int_{0}^{\infty}\frac{1}{x}\left(1-\left(e^{x}-1\right)e^{-x}-\frac{x}{e^{x}-1}+xe^{-x}\right)dx\\ & =\int_{0}^{\infty}\frac{1}{x}\left(-\frac{x}{e^{x}-1}+\left(1+x\right)e^{-x}\right)dx\\ & =\int_{0}^{\infty}\left(-\frac{1}{e^{x}-1}+x^{-1}e^{-x}+e^{-x}\right)dx\\ & =\int_{0}^{\infty}\left(-\frac{1}{e^{x}-1}+x^{-1}e^{-x}+\frac{1-e^{-x}}{e^{x}-1}\right)dx\\ & =\int_{0}^{\infty}\left(x^{-1}e^{-x}-\frac{e^{-x}}{e^{x}-1}\right)dx\\ & =\int_{0}^{\infty}\left(x^{-1}e^{-x}-\frac{e^{-2x}}{1-e^{-x}}\right)dx\\ & =\left[\log\left(x\right)e^{-x}\right]_{0}^{\infty}+\int_{0}^{\infty}\left(\log\left(x\right)e^{-x}-e^{-2x}\sum_{k=0}^{\infty}e^{-kx}\right)dx\\ & =-\lim_{x\rightarrow0}\log\left(x\right)+\left[\frac{d}{dt}\int_{0}^{\infty}x^{t}e^{-x}dx\right]_{t=0}-\int_{0}^{\infty}\sum_{k=0}^{\infty}e^{-\left(k+2\right)x}dx\\ & =-\lim_{x\rightarrow1}\log\left(1-x\right)+\left[\frac{d}{dt}\Gamma\left(t+1\right)\right]_{t=0}+\sum_{k=0}^{\infty}\frac{1}{k+2}\left[e^{-\left(k+2\right)x}\right]_{0}^{\infty}\cmt{\text{積分と総和の順序変更}}\\ & =\lim_{x\rightarrow1}\sum_{k=1}^{\infty}\frac{x^{k}}{k}+\left[\Gamma\left(t+1\right)\frac{d}{dt}\log\Gamma\left(t+1\right)\right]_{t=0}-\sum_{k=0}^{\infty}\frac{1}{k+2}\\ & =\sum_{k=1}^{\infty}\frac{1}{k}+\left[\Gamma\left(t+1\right)\psi\left(t+1\right)\right]_{t=0}-\sum_{k=1}^{\infty}\frac{1}{k+1}\\ & =\sum_{k=1}^{\infty}\frac{1}{k}+\Gamma\left(1\right)\psi\left(1\right)+H_{1}-\sum_{k=1}^{\infty}\frac{1}{k}\\ & =\Gamma\left(1\right)\psi\left(1\right)+H_{1}\\ & =1-\gamma \end{align*}