How to write an "index set" Imagine I have a finite sequence $\{s_i\}_{1\leq i\leq N}$, for some $N$. Now assume I want to sum all the terms of such sequence apart from the term $s_i$, for some fixed $i$. One way to do it is to simply write
$$
S=\sum_{j\neq i}s_j
$$
However, I want to explicitly write it by defining a $k$ in some set $K$ such that
$$
S=\sum_{k\in K}s_{i+k}
$$
Would the set $K=(\mathbb{Z}/N)\setminus\{0\}$ work? Any better way of writing it?
 A: The most obvious (and readable) is to write
\begin{align}
\sum_{j\neq i}s_j.
\end{align}
Once in a while, this notation can become ambiguous, because the reader might interpret the sum to be over all pairs of $(i,j)$ such that $j\neq i$ (in this specific case, it's next to impossible to misinterpret though). To be slightly more explicit, you can write
\begin{align}
\sum_{\substack{1\leq j\leq N\\
j\neq i}}s_j
\end{align}
If you want to write sets, then I'd suggest defining $J=\{j\in \Bbb{N}\,: 1\leq j\leq N\quad\text{and} \quad j\neq i\}$. Then you can write your sum as
\begin{align}
\sum_{j\in J}s_j
\end{align}
Now, if you want to write it in the form $s_{i+k}$, then you have to define the set $K=\{k\in \Bbb{Z}\,: \text{$1-i\leq k\leq N-i$ and  $k\neq 0$}\}$, and then you can write the sum as
\begin{align}
\sum_{k\in K}s_{i+k},
\end{align}
but honestly this is just a confusing way of writing things.
What you could do though is define a new sequence of numbers $\{\sigma_n\}_{n\in\Bbb{Z}}$, such that $\sigma_n=s_n$ if $1\leq n\leq N$, and $\sigma_n=0$ otherwise. Then you can write the sum in question as
\begin{align}
\sum_{k\in \Bbb{Z}\setminus \{0\}}\sigma_{i+k}.
\end{align}
This simplifies the index set, but slightly complicates the sequence.
