sequences and series - Questions on two Formulas for $zeta(s)$

This question is related to the following two formulas for $\zeta(s)$.

(1) $\quad\zeta(s)=\frac{1}{1-2^{1-s}}\sum\limits_{n=0}^\infty\frac{1}{2^{n+1}}\sum\limits_{k=0}^n\frac{(-1)^k\binom{n}{k}}{(k+1)^s},\quad s\ne 1\quad\text{(see ref(1) and formula (21) at ref(2))}$

(2) $\quad\zeta(s)=\frac{1}{s-1}\sum\limits_{n=0}^\infty\frac{1}{n+1}\sum\limits_{k=0}^n\frac{(-1)^k\binom{n}{k}}{(k+1)^{s-1}}\qquad\qquad\qquad\text{(see ref(1) and formula (22) at ref(2))}$

Formula (1) above is claimed to converge for $s\ne 1$ at ref(2), but note that $\frac{1}{1-2^{1-s}}$ exhibits a complex infinity at $s=1+i\frac{2\,\pi\,j}{\log(2)}$ where $j\in \mathbb{Z}$ which seems consistent with the convergence claim at ref(1).

Question (1): Is it true that formula (1) converges for $s\ne 1+i\frac{2\,\pi\,j}{\log(2)}$ where $j\in \mathbb{Z}$ versus $s\ne 1$? Or is there an argument about zeros and poles cancelling each other out when formula (1) for $\zeta(s)$ is evaluated at $s=1+i\frac{2\,\pi\,j}{\log(2)}$ where $j\in \mathbb{Z}$ similar to the argument for the convergence of the right side of the functional equation $\zeta(s)=2^s π^{s−1}\sin\left(\frac{π\,s}{2}\right)\,\Gamma(1−s)\,\zeta(1−s)$ at positive integer values of s (e.g. see Using the functional equation of the Zeta function to compute positive integer values)?

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Since originally posting question (1) above, I discovered the following Wikipedia article which I believe provides some insight.

Wikipedia Article: Landau's problem with $\zeta(s)=\frac{\eta(s)}{0}$ and solutions

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Formula (2) above is claimed to be globally convergent, but seems to exhibit a significant divergence (see Figure (1) below).

Question (2): Is there an error in formula (2), or is there a conditional convergence requirement associated with formula (2) when the outer series is evaluated for a finite number of terms?

ref(1): Wikipedia Article: Riemann zeta function, Representations, Globally convergent series

ref(2): Sondow, Jonathan and Weisstein, Eric W. "Riemann Zeta Function." From MathWorld--A Wolfram Web Resource.

12/10/2018 Update:

I'm now wondering if formula (2) for $\zeta(s)$ is perhaps only valid for $s\in\mathbb{Z}$.

The following plot illustrates formula (2) for $\zeta(s)$ evaluated for the first $100$ terms.

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Figure (1): Illustration of Formula (2) for $\zeta(s)$

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The following discrete plot illustrates formula (2) for $\zeta(s)$ minus $\zeta(s)$ where formula (2) is evaluated for the first $100$ terms in blue and the first $1000$ terms in orange.

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Figure (2): Discrete Plot of Formula (2) for $\zeta(s)$ minus $\zeta(s)$

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