The Riemann Zeta Function can be defined as: $\zeta(s)=\sum \frac 1 {n^s}$ for $s>1$. The series converges for $s>1$. wiki (https://en.wikipedia.org/wiki/Riemann_zeta_function) mentions that:
An extension of the area of convergence can be obtained by rearranging the original series.
(1) The series $\zeta (s)={\frac {1}{s-1}}\sum _{n=1}^{\infty }\left({\frac {n}{(n+1)^{s}}}-{\frac {n-s}{n^{s}}}\right)$ converges for $Re(s) > 0$
(2) while $\zeta (s)={\frac {1}{s-1}}\sum _{n=1}^{\infty }{\frac {n(n+1)}{2}}\left({\frac {2n+3+s}{(n+1)^{s+2}}}-{\frac {2n-1-s}{n^{s+2}}}\right)$ converges even for $Re(s) > −1$. In this way, the area of convergence can be extended to Re(s) > −k for any negative integer $−k$.
Query 1: I am very new to analysis so can someone please explain this series rearrangement process for a layman?
Query 2: Using (1) I tried to extract the real and imaginary parts of the $\zeta (s)$ as:
$Re(\zeta (s))={\frac {1}{s-1}}\sum _{n=1}^{\infty}{\frac {n(C_{n+1})}{(n+1)^{b}}} - {\frac {(n-b)C_n}{n^{b}}}+{\frac {cS_n}{n^{b}}}$
$Img(\zeta (s))={\frac {i}{s-1}}\sum _{n=1}^{\infty}-{\frac {n(S_{n+1})}{(n+1)^{b}}} + {\frac {(n-b)S_n}{n^{b}}}+{\frac {cC_n}{n^{b}}}$
where $C_{n}= \cos(c \ln(n)), S_{n}= \sin(c \ln(n)), C_{n+1}= \cos(c \ln(n+1)), S_{n+1}= \sin(c \ln(n+1))$ and $[s=b+ic]$.
I am unsure if these are correct (i.e. these converge for $Re(s)>0$). Can someone please clarify what is incorrect here?