Confusion on the proof of the Divergence Test
Solution 1:
I think you just have a notational confusion: the notation $\sum_{k = n}^n a_k$ literally just means the sum of all $a_k$ where $k$ starts at $n$ and increasing by $1$ until $k$ gets to $n$. That is, only a single term, $a_n$ itself! So $\sum_{k = n}^n a_k$ is a sum of one term only, and is equal to $a_n$ on the nose. Thus we can go from from $\lvert\sum_{k=n}^{n}a_k \rvert < \epsilon$ to $\lvert a_n\rvert<\epsilon$ because they mean the same thing!