Why is the limit of an infinite series the same as limit of sequence of partial sums?
Context: I was solving a question to find the limit of an infinite series (real analysis). One method stated that it is equal to the limit of sequence of partial sums. I did not understand why.
Edit: sequence of partial sums
The value of a series (any series is generally defined as an addition of infinite terms, otherwise it will be a sum) is generally defined as the value of the limit (if exists) of the sequence defined by it partial sums, that is
$$\sum_{k=m}^\infty a_k:=\lim_{n\to\infty}\sum_{k=m}^n a_k$$
Many mathematicians use the symbol "$:=$" to mean: the left hand side is defined to represent the right hand side.
Here $\sum_{k=m}^n a_k$ is a partial sum because the series is partially added up to just the $n$-th term.
If we set $x_n:=\sum_{k=m}^n a_k$ (observe the use of the symbol "$:=$") then the value of the series is equivalently defined to be the value (when it exists) of the sequence $\{x_n\}_{n\in\Bbb N}$[*].
[*] If $m> 0$ (if the series start with an index distinct of zero) then the sums of the kind $\sum_{k=m}^na_k$ are defined to have value zero when $n<m$. This is a convention called the empty sum.