A series ∑ aₙ converges to S if the sequence of partial sums Sₙ = a₁ + a₂ + ... + aₙ converges to S using epsilon-N. A series converges if and only if it is Cauchy: for every ε > 0, there exists N such that for all n > m ≥ N, |Sₙ − Sₘ| < ε. This provides a rigorous foundation for all convergence tests.
From your prerequisite on epsilon-N convergence, you have a precise definition of what it means for a sequence to converge: (aₙ) converges to L if for every ε > 0, there exists N such that for all n ≥ N, |aₙ − L| < ε. The key insight for rigorous series convergence is that a series is not a new object — it is a sequence. Specifically, the series ∑ aₙ is defined as the sequence of partial sums S₁ = a₁, S₂ = a₁ + a₂, S₃ = a₁ + a₂ + a₃, and so on. Saying the series converges to S means the sequence (Sₙ) converges to S in the exact epsilon-N sense you already know. Every tool you developed for sequences applies immediately to series, through this translation.
Your prerequisite on convergence tests gave you practical criteria: ratio test, integral test, comparison test, alternating series test. Each of these can now be understood as a sufficient condition for the partial sum sequence (Sₙ) to be convergent. The ratio test, for example, works by showing the terms decrease fast enough that the partial sums form a Cauchy sequence — the "tail" of the series gets arbitrarily small. The Cauchy criterion for series makes this precise: ∑ aₙ converges if and only if for every ε > 0 there exists N such that for all n > m ≥ N, |aₘ₊₁ + aₘ₊₂ + … + aₙ| < ε. This says that the sum of any sufficiently "late" block of terms is small, which is exactly the condition that the partial sums form a Cauchy sequence.
A critical consequence of the Cauchy criterion is the necessary condition for convergence: if ∑ aₙ converges, then aₙ → 0. Proof by contradiction using Cauchy: if aₙ does not approach 0, we can find blocks of single terms with |aₙ| ≥ ε for infinitely many n, which violates the Cauchy condition for m = n − 1. Note carefully that this gives a *necessary* condition, not a sufficient one. The harmonic series ∑ 1/n has terms going to 0, yet the partial sums grow without bound — divergence despite the necessary condition being met. The Cauchy criterion catches this: the block 1/(N+1) + 1/(N+2) + … + 1/(2N) is always at least 1/2, no matter how large N is.
The rigorous framework also distinguishes absolute convergence (∑ |aₙ| converges) from ordinary convergence (∑ aₙ converges). This distinction, which is the next topic in the builds-toward list, becomes cleaner in the rigorous setting: absolutely convergent series converge regardless of how their terms are rearranged, while conditionally convergent series (convergent but not absolutely) can be rearranged to converge to any target value or to diverge. The Cauchy criterion makes this precise — absolute convergence means the partial sums of |aₙ| are Cauchy, which provides stronger control over rearrangements than ordinary Cauchy behavior alone. Every convergence test from your earlier study can be reinterpreted as establishing absolute or conditional convergence, with the rigorous epsilon-N framework supplying the proofs that the tests themselves only hinted at.