$\textbf{Cauchy's convergence test}$
The series $\sum_{k=1}^{\infty}a_k$ is convergent if and only if for every $\varepsilon>0$ there exists $N\in \N$ such that for all $m,n\in \N$ $$n\geq m\geq N \implies \Bigg\lvert\sum_{k=m}^na_k\Bigg\rvert<\varepsilon.$$
$\textbf{Cauchy's convergence test}$
The series $\sum_{k=1}^{\infty}a_k$ is convergent if and only if for every $\varepsilon>0$ there exists $N\in \N$ such that for all $m,n\in \N$ $$n\geq m\geq N \implies \Bigg\lvert\sum_{k=m}^na_k\Bigg\rvert<\varepsilon.$$
copied
𝐂𝐚𝐮𝐜𝐡𝐲’𝐬 𝐜𝐨𝐧𝐯𝐞𝐫𝐠𝐞𝐧𝐜𝐞 𝐭𝐞𝐬𝐭
The series ∑ₖ₌₁ ᪲ 𝑎ₖ is convergent if and only if for every 𝜀>0 there exists 𝑁∈ℕ such that for all 𝑚,𝑛∈ℕ
𝑛≥𝑚≥𝑁 ⟹ ∣∑ₖ₌ₘⁿ𝑎ₖ∣<𝜀.