This question is related to the function $f_a(s)$ defined in formula (1) below with the infinite series representation defined in formula (2) below where $a$ is a positive integer. Note that $a=1$ corresponds to the Dirichlet eta function $\eta(s)$, and $a=2$ corresponds to the Dirichlet beta function $\beta(s)$.
(1) $\quad f_a(s)=(2\ a)^{-s} \left(\zeta\left(s,\frac{1}{2\ a}\right)-\zeta\left(s,\frac{1}{2\ a}+\frac{1}{2}\right)\right)$
(2) $\quad f_a(s)=\underset{K\to\infty}{\text{lim}}\left(\sum\limits_{n=0}^K \frac{(-1)^n}{(a\ n+1)^s}\right)\ ,\quad\Re(s)>0$
I believe the infinite series defined in formula (3) below is valid for all $s$, and the finite series defined in formula (4) below is valid at $s=-K$ where $K$ is a non-negative integer. Formulas (3) and (4) below are based on a conjectured formula for the Dirichlet eta function $\eta(s)$ which I previously defined in formula (1) of this question.
(3) $\quad f_a(s)=\underset{K\to\infty}{\text{lim}}\left(\frac{1}{2^{K+1}}\sum\limits_{n=0}^K \frac{(-1)^n }{(a\ n+1)^s}\sum\limits_{k=0}^{K-n} \binom{K+1}{K-n-k}\right)$
(4) $\quad f_a(-K)=\frac{1}{2^{K+1}}\sum\limits_{n=0}^K (-1)^n (a\ n+1)^K \sum\limits_{k=0}^{K-n} \binom{K+1}{K-n-k}\ ,\quad K\in\mathbb{Z}\land K\ge 0$
I believe formula (3) above is exactly equivalent to formula (5) below for all values of $s$, all non-negative integer values of $K$, and all positive integer values of $a$; and also formula (4) above is exactly equivalent to formula (6) below. Formulas (5) and (6) below are based on this well-known formula for the Dirichlet eta function $\eta(s)$.
(5) $\quad f_a(s)=\underset{K\to\infty}{\text{lim}}\left(\sum\limits_{n=0}^K \frac{1}{2^{n+1}}\sum\limits_{k=0}^n \frac{(-1)^k\ \binom{n}{k}}{(a\ k+1)^s}\right)$
(6) $\quad f_a(-K)=\sum\limits_{n=0}^K \frac{1}{2^{n+1}}\sum\limits_{k=0}^n (-1)^k\ \binom{n}{k}\ (a\ k+1)^K\ ,\quad K\in\mathbb{Z}\land K\ge 0$
Note formulas (3) and (4) above have the exponentiation operation in the outer sum whereas formulas (5) and (6) above have the exponentiation operation in the inner nested sum which leads to formulas (3) and (4) having fewer exponentiation operations and faster evaluation times.
Question: Is it true formula (3) above is a globally convergent infinite series representation of the function $f_a(s)$ defined in formula (1) above for all positive integer values of $a$, and formula (3) above is exactly equivalent to formula (5) above for all values of $s$, all integer values of $K\ge 0$, and all integer values of $a>0$?
Side-note:
I've read arithmetic statements can be encoded in terms of Binomial coefficients (see Encodings of arithmetic (by different operations)). This observation along with Godel's first incompleteness theorem seems to imply there may be some true statements involving Binomial coefficients that are impossible to prove, and I'm wondering whether the equivalence of formulas (3) and (5) above is perhaps an example.
The following figures illustrate formula (3) above for $f_a(s)$ for several values of $a$ in orange overlaid on the reference function $f_a(s)$ defined in formula (1) above in blue where formula (3) is evaluated at $K=100$.
Figure (1): Illustration of formula (3) for $f_1(s)=\eta(s)$ in orange overlaid on the blue reference function
Figure (2): Illustration of formula (3) for $f_2(s)=\beta(s)$ in orange overlaid on the blue reference function
Figure (3): Illustration of formula (3) for $f_3(s)=6^{-s} \left(\zeta \left(s,\frac{1}{6}\right)-\zeta \left(s,\frac{2}{3}\right)\right)$ in orange overlaid on the blue reference function