This is the key reframing of rigorous series convergence: a series is not a new kind of object. It is defined as the sequence of its partial sums. Saying ∑ aₙ converges to S means the sequence (Sₙ) converges to S in the exact epsilon-N sense. Every tool developed for sequences — including the Cauchy criterion — applies directly to series through this identification.
Question 2 Multiple Choice
A student argues: 'Since 1/n → 0, the series ∑ 1/n must converge.' What is the precise error?
AThe terms 1/n do not actually approach 0 — they approach 1
Baₙ → 0 is necessary for convergence but not sufficient; the harmonic series diverges despite its terms going to 0
CThe ratio test overrides all necessary condition arguments
D1/n is not a valid series term because it is unbounded above
The condition aₙ → 0 is necessary for convergence: if a series converges, its terms must go to 0 (provable directly from the Cauchy criterion). But necessity is not sufficiency. The harmonic series ∑ 1/n is the canonical counterexample — its terms go to 0, yet the partial sums grow without bound. The Cauchy criterion exposes why: the block 1/(N+1) + 1/(N+2) + ⋯ + 1/(2N) ≥ 1/2 for every N, so the partial sums can never become and stay within ε of each other.
Question 3 True / False
If the series ∑ aₙ converges, then the terms aₙ must approach 0.
TTrue
FFalse
Answer: True
This follows directly from the Cauchy criterion applied to single-term blocks. If the partial sums form a Cauchy sequence, then for any ε > 0, there exists N such that |Sₙ − Sₙ₋₁| < ε for all n ≥ N. But |Sₙ − Sₙ₋₁| = |aₙ|. So aₙ → 0. This is a necessary condition — its failure (aₙ not going to 0) immediately implies divergence via the divergence test.
Question 4 True / False
If the terms aₙ → 0, then the series ∑ aₙ converges.
TTrue
FFalse
Answer: False
This is the most important false statement in series theory. The harmonic series ∑ 1/n is the definitive counterexample: 1/n → 0, yet the series diverges. The Cauchy criterion for series requires that arbitrarily late *blocks* of consecutive terms sum to something small — not just that individual terms become small. The harmonic series fails this: no matter how far out you go, you can always find a block of terms (from N+1 to 2N) summing to at least 1/2.
Question 5 Short Answer
Explain how the Cauchy criterion for series follows from the Cauchy criterion for sequences, and state what it says about convergence.
Think about your answer, then reveal below.
Model answer: A series ∑ aₙ is defined as the sequence of partial sums (Sₙ). A sequence converges if and only if it is Cauchy. So ∑ aₙ converges if and only if (Sₙ) is Cauchy: for every ε > 0, there exists N such that |Sₙ − Sₘ| < ε for all n > m ≥ N. But |Sₙ − Sₘ| = |aₘ₊₁ + aₘ₊₂ + ⋯ + aₙ| — the sum of a block of consecutive terms. So convergence requires that all sufficiently late blocks of consecutive terms have arbitrarily small sum.
This criterion gives the rigorous foundation for all convergence tests. The ratio test, for instance, works by showing the terms shrink geometrically fast enough that late blocks get arbitrarily small — exactly the Cauchy condition. The harmonic series fails because no block sum ever gets below 1/2.