Consider a sequence $\{a_i\}_{i\in \mathbb{N}}$. I would like your help to understand the relation among
(1) $\sum_{i=1}^{\infty} a_i$
(2) $\lim_{n\rightarrow \infty} \sum_{i=1}^n a_i$
(3) Riemann integral
(4) Integrating over an uncountably infinite set
More context:
I am asking this because I have derived some results that hold when $\sum_{i=1}^n a_i$ enters my problem. The professor asked me to extend those results to the infinite case and I don't know whether (A) I should replace $\sum_{i=1}^n a_i$ with $\sum_{i=1}^{\infty} a_i$, (B) I should replace $\sum_{i=1}^n a_i$ with $\lim_{n\rightarrow \infty}\sum_{i=1}^{n} a_i$, (C) I should replace $i\in \{1,2,...,n\}$ with $i\in \mathcal{I}$ and $\mathcal{I}$ uncountably infinite (and then I replace the sum with an integral?). I'm a beginner of asymptotics.
So, usually, the definition of $1$ is $2$, i.e the infinite sum is defined as the limit of the finite sums.
The Riemann integral is defined as the limit of finite sums. Sums over uncountable sets are not defined. Specifically, if you have an interval $[a,b]$, you chop it up/partition it using the set $\{a =x_0, \cdots , x_n = b\}$. then you pick points $t_i$ between each $x_i,x_{i+1}$. You then define Riemann sums
$$ \sum_{i = 0}^{n} f(t_i) (x_{i+1} - x_i) $$ for any partition. You take the limit then as the mesh of the partition approaches $0$. By the mesh, we mean the maximum distance between any two points in the partition, i.e $\mathrm{mesh} (P) = \max_{0 \leq i \leq n-1}(x_{i+1} - x_i)$. Hopefully this makes some sense.