Help with series convergence and divergence concept

Question

When you are trying to figure out whether a series converges or diverges why can you test for convergence at any term in the series as opposed to having to start at the beginning of the series. Is it because the ratio is the same throughout the series.

Answer 1

Write $S_n:= \sum_{1 \leq k \leq n} a_k$ and $T_n := \sum_{k > n} a_k$. Write $S:= \sum_{k=1}^{\infty} a_k$. Can you show the following?

$S$ exists iff $T_n$ exists for all $n \in \mathbb{N}$, and in that case we have $S = S_n + T_n$.