Motivated by Gautschi double inequality, $$ \frac{n^{s}}{n^{\small1}}\ge\frac{\Gamma(n+s)}{\Gamma(n+1)}\ge\frac{(n+1)^{s}}{(n+1)^{\small1}}\ge\frac{\Gamma(n+1+s)}{\Gamma(n+1+1)}\ge\,\cdots \quad\colon\,0\lt{s}\lt1\tag{1} $$ From the main definition of zeta function, $$ \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}\,\,\,\colon\,Re\{s\}\gt1 \space\Rightarrow\space \zeta(1-s)=\sum_{n=1}^{\infty}\frac{n^{s}}{n^{\small1}} \qquad\colon\,Re\{s\}\lt0\tag{2} $$ And the sum identity of gamma function, $$ \sum_{n=0}^{\infty}\frac{\Gamma(n+s)}{n!}=0 \quad\Rightarrow\quad \Gamma(s)=-\sum_{n=1}^{\infty}\frac{\Gamma(n+s)}{\Gamma(n+1)} \qquad\colon\,Re\{s\}\lt0\tag{3} $$
How to Prove, Disprove, or Justify: $$ \zeta(1-s)+\Gamma(s)=\sum_{n=1}^{\infty}\left[\,\frac{n^{s}}{n^{\small1}}-\frac{\Gamma(n+s)}{\Gamma(n+1)}\,\right] \qquad\qquad\colon\,0\lt{s}\lt1\tag{4} $$ Does it hold if extended to complex plan $\,s\in\mathbb C\,$ inside the critical strip $\small 0\lt Re\{s\}\lt1\,$ ?
“By the monotonic decreasing behavior of the inequality, the subtracting result of the two divergent series converge one step forward, covering the critical strip!”
My first thought is to give an integral representation for the general term of the series, $$\frac{1}{n^{1-s}}-\frac{B(n+s,1-s)}{\Gamma(1-s)}=\frac{1}{\Gamma(1-s)}\left(\int_{0}^{+\infty}x^{-s}e^{-n x}\,dx-\int_{0}^{1}x^{-s}(1-x)^{n+s-1}\,dx\right)$$
$$\frac{1}{n^{1-s}}-\frac{B(n+s,1-s)}{\Gamma(1-s)}=\frac{1}{\Gamma(1-s)}\left(\int_{0}^{+\infty}x^{-s}e^{-n x}\,dx-\int_{0}^{+\infty}(1-e^{-x})^{-s}e^{-(n+s)x}\,dx\right)$$ Summing over $n\geq 1$ we get:
$$\sum_{n\geq 1}\left(\frac{n^s}{n^1}-\frac{\Gamma(n+s)}{\Gamma(n+1)}\right)=\frac{1}{\Gamma(1-s)}\int_{0}^{+\infty}\left(\frac{1}{x^s}-\frac{1}{(e^x-1)^s}\right)\frac{dx}{e^x-1}$$ for every $s$ with real part $\in(0,1)$. The explicit computation of the last integral as $\,\frac{\pi}{\sin(\pi s)}+\Gamma(1-s)\,\zeta(1-s)\,$ proves OP's identity $(4)$. For the computation we may use, for instance, the classical application of Ramanujan's master theorem to Bernoulli polynomials.