First order approximation of $\zeta$(s) at s=1

Question

I was playing around with Wolfram Alpha. I found one interesting thing when I asked it to evaluate this particular summation.$$\Sigma_{n=1}^\infty\frac{1}{n^{1+10^{- 10}}}$$ It returned this$$ \approx10^{10}+\gamma$$ That's right the number itself plus the Euler–Mascheroni constant. I tried it for various other values and it turned out to follow the same pattern. As the value got smaller the closer it followed that pattern. Image of Wolfram Alpha result click here So formally stating $$ \lim _{s \rightarrow0 } \zeta(1+s) = 1/s+ \gamma$$ Is this true? And why is this true? Could it be proven by taking the first order approximation of the zeta function.

Answer 1

$$\sum_{n=1}^\infty\frac{1}{n^{1+\epsilon}}=\zeta (1+\epsilon)=\frac{1}{\epsilon }+\gamma -\gamma _1 \epsilon +\frac{\gamma _2 }{2}\epsilon ^2+O\left(\epsilon ^3\right)$$ where appear the generalized Stieltjes constants.