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Question: Is it true that formula (1) below for $\eta(s)$ converges for $\Re(s)>-m$ as $N\to\infty$ and if so, is formula (1) somehow related to the derivation of formula (2) below for $\eta(s)$ which is globally convergent?

(1) $\quad\eta_{\,m,N}(s)=\sum\limits_{n=1}^N\frac{(-1)^{n-1}}{n^s}+\frac{1}{2^m}\sum\limits_{n=1}^m\left(\sum\limits_{i=1}^{m-n+1}\binom{m}{m-n+1-i}\right)\frac{(-1)^{N+n-1}}{(N+n)^s}\,,\\$ $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad Re(s)>-m?\land N\to\infty$

(2) $\quad\eta(s)=\sum\limits_{n=0}^\infty\frac{1}{2^{n+1}}\sum\limits_{k=0}^n (-1)^k\,\binom{n}{k}\frac{1}{(k+1)^s}\quad$(see Dirichlet eta function: Numerical Algorithms)


The following figures illustrate formula (1) for $\eta(s)$ above seems to converge for $\Re(s)>-m$ as $N\to\infty$ when evaluated with $m\in\{1,2,3,4\}$. In the following figures formula (1) is 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.


The red discrete dot at $s=-m$ visible in most of the figures below illustrates $\frac{1}{2}\left(\eta_{\,m,100}(-m)+\eta_{\,m,101}(-m)\right)$. I'll note that I believe $\frac{1}{2}\left(\eta_{\,m,N}(-m)+\eta_{\,m,N+1}(-m)\right)$ evaluates exactly to $\eta(-m)$ for all values of $N\ge 0$, and $\eta_{\,m,N}(-j)$ evaluates exactly to $\eta(-j)$ where $j\in \mathbb{Z}\land 0\le j<m$ for all values of $N\ge 0$.


This leads to the following representations for $\eta(-j)$ which I believe are valid for $j\in\mathbb{Z}\land j\geq 0$ where $\binom{a}{b}$ is the binomial coefficient and $(a)_k$ is the Pochhammer symbol.

(3) $\quad\eta(-j)=\frac{1}{2^{j+1}}\sum\limits_{n=1}^{j+1}(-1)^{n-1}\,n^j\sum\limits_{i=1}^{j+2-n}\binom{j+1}{j+2-n-i}\\$ $\qquad\qquad\quad\,\,=\frac{1}{2^{j+1}}\sum\limits_{n=1}^{j+1}(-1)^{n-1}\,n^j\binom{j+1}{j+1-n}\,_2F_1(1,n-j-1;n+1;-1)\\ $ $\qquad\qquad\quad\,\,=\frac{1}{2^{j+1}}\sum\limits_{n=1}^{j+1}(-1)^{n-1}\,n^j\binom{j+1}{j+1-n}\sum\limits_{k=0}^{\infty}\frac{(-1)^k(n-j-1)_k}{(n+1)_k}$


Figure (1a) Figure (1b)

Figure (1): Illustration of Formula (1) for $\eta(s)$ with $m=1$


Figure (2a) Figure (2b)

Figure (2): Illustration of Formula (1) for $\eta(s)$ with $m=2$


Figure (3a) Figure (3b)

Figure (3): Illustration of Formula (1) for $\eta(s)$ with $m=3$


Figure (4a) Figure (4b)

Figure (4): Illustration of Formula (1) for $\eta(s)$ with $m=4$


The following table illustrates $f(m,n)=\sum\limits_{i=1}^{m-n+1}\binom{m}{m-n+1-i}$ for $m=1..n$ which seems to correlate to https://oeis.org/A104709.


$\begin{array}{cc} \text{n} & \text{f(m,n)} \\ 1 & \{1\} \\ 2 & \{3,1\} \\ 3 & \{7,4,1\} \\ 4 & \{15,11,5,1\} \\ 5 & \{31,26,16,6,1\} \\ 6 & \{63,57,42,22,7,1\} \\ 7 & \{127,120,99,64,29,8,1\} \\ 8 & \{255,247,219,163,93,37,9,1\} \\ 9 & \{511,502,466,382,256,130,46,10,1\} \\ 10 & \{1023,1013,968,848,638,386,176,56,11,1\} \\ \end{array}$

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    $\begingroup$ Forjmula (2) comes from MSE question 3182855 "Question on convergence of formula for Dirichlet eta function $\eta(s)$". $\endgroup$
    – Somos
    Commented Jun 7, 2019 at 1:15

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