On a representation of the Riemann zeta function

In this paper a new representation of the Riemann zeta function in the disc $U(2,1)$ is obtained: $\zeta (z) = \dfrac{1}{z-1} + \displaystyle\sum_{n=0}^\infty (-1)^n\alpha_n(z-2)^n,$ where the coefficients $\alpha_k$ are real numbers tending to zero. Hence is obtained $\gamma=\displaystyle\lim_{m\rightarrow\infty} \left[\displaystyle\sum_{k=0}^{n-1} \dfrac{\zeta^{(k)}(2)}{k!}-n\right]$, where $\gamma$ is the Euler–Mascheroni constant.