- let $f:\Bbb{N}\to \Bbb{R}$ a sequence, $n \in \Bbb{N}$ and $S \in \Bbb{R}$, $S$ is $n$th-partial sum of $f$ if $$S=\sum_{i=0}^nf(i)$$
- let $g:\Bbb{N} \to \Bbb{R}$ and $h:\Bbb{N} \to \Bbb{R}$ two sequences, $g$ is series associated to $h$, $g:=\sum_{i=0}^\infty h(i)$, if $$\forall x \in \Bbb{N}(g(x)=\sum_{i=0}^xh(i))$$
Is it correct? Thanks in advance!
This is a point of definitions and naming conventions. My preferred definition is that when $$ S_n = \sum_{i=0}^n a_i $$ we say that the sequence $S_n$ is the series of term $a_n$. Both $S_n$ and $a_n$ are sequences. While $$ S = \lim_{n\to \infty} S_n = \sum_{i=0}^\infty a_i $$ is the sum of the series of term $a_n$. $S$ is a number (or $\pm \infty$).