Question: Is it true that for a zero-mean q-periodic Dirichlet series (where $a_n=a_{n+q}$ and $\sum_{n=1}^q a_n=0$) formula (2) below extends the range of convergence of formula (1) below from $\Re(s)>0$ to $\Re(s)>-1$ as $N\to\infty$?
(1) $\quad f(s)=\sum_\limits{n=1}^N a_n\,n^{-s}\,,\quad\Re(s)>0\land N\to\infty$
(2) $\quad f(s)=\frac{1}{q}\sum\limits_{m=0}^{q-1}\sum\limits_{n=1}^{N+m} a_n\,n^{-s}\\$$\qquad\qquad\,\,\,=\sum\limits_{n=1}^N a_n\,n^{-s}+\frac{1}{q}\sum\limits_{n=1}^{q-1}(q-n)\,a_{N+n}\,(N+n)^{-s}\,,\quad\Re(s)>-1?\land N\to\infty$
As the simplest example, the following figures illustrate formulas (1) and (2) above for the Dirichlet eta function $\eta(s)=(1-2^{1-s})\,\zeta(s)$ evaluated with upper limits $N=100$, $N=101$, $N=1000$, and $N=1001$ in orange, green, red, and purple respectively overlaid on the blue reference function. Note formula (2) seems to converge in the interval $-1<x<0$ (see figure (4) below) in a manner analogous to the convergence of formula (1) in the interval $0<x<1$ (see figure (2) below). Note for $\eta(s)$, $a_n=(-1)^{n-1}$ and $q=2$.
Figure (1): Illustration of Formula (1) for $\eta(s)$
Figure (2): Illustration of Formula (1) for $\eta(s)$
Figure (3): Illustration of Formula (2) for $\eta(s)$
Figure (4): Illustration of Formula (2) for $\eta(s)$