My version of definition of series is :
If $X=(x_n)$ is a sequence in $\Bbb R$ then the series generated by $X$ is the sequence $S=(s_n)$ defined as
$$s_1=x_1,$$ $$s_2=s_1+x_2,$$ $$s_3=s_2+x_3,$$ $$...$$ $$s_k=s_{k-1}+x_k,$$
and so on. The numbers $x_n$ are called the $n-th$ term of the series and the numbers $s_n$ are called the partial sums of the series. If the sequence $S$ converges at a point (in $\Bbb R$) then, we say, this seies is convergent, and call this limit the sum of the series. If this limit does not exist, we call the series to be diverging.
It is convenient to use the symbol such as $$\sum x_n\tag 1$$ to denote the infinite series $S,$ generated by the sequence $X=(x_n)$ and also denote the value, $\lim S$ in case the limit exists. Thus, the symbol $(1)$ above, is merely "a way of exhibiting" an infinite series whose convergence or divergence is to be tested.
(This definition is rather a popular one.)
In short, I regard,
"A series is a special type of sequence generated by a sequence."
The problem arised, when I was going through some tests related to convergence of a series, for example: Comparison Test of First Kind, Limit Test, D'Alambert's ratio Test, etc. Say for example, in the Comparison Test was given as:
Let $ \sum u_n$ and $\sum v_n$ be two series of positive real numbers and there is a natural number $m$ such that $u_n\leq kv_n,\forall n\geq m,$ $k$ being a fixed positive number. Then, $\sum u_n$ is convergent if $\sum v_n$ is divergent.
But the part of the statement, where it said, "Let $ \sum u_n$ and $\sum v_n$ be two series of positive real numbers" (at the beginning) seems ambiguous to me. This is because, I really don't know what is meant by the phrase "series of positive real numbers". Does the phrase, "series of positive real numbers" means that the series $\sum u_n$ and $\sum v_n$ that are generated by the respective sequences $(u_n)$ and $(v_n)$ has all terms in the sequence,i.e $(u_n)$ or $(v_n)$ positive ? (In other words, does the phrase implies $u_n,v_n\gt 0,\forall n\in\Bbb N$ ?) Or is it just they mean that the each of the terms in the sequence of partial sums of each (given) series are positive ?
A “series of blah numbers” means a series where each of its terms has property blah. It need not be true that the partial sums have property blah too.
I think titling your post “A dilemma” is a bit strong. Just “A question regarding$\ldots$” would be fine.