Convergence and Divergence of Sequences: Notes and FAQs

A sequence converges when its terms approach one fixed number L and stay there; it diverges in every other case — blowing up to infinity, oscillating between values, or wandering without pattern. Classifying a sequence correctly is the first step in any limit argument.

This matters because only convergent sequences give trustworthy approximations. Knowing the standard tests lets you decide convergence at a glance instead of wrestling with ε–N definitions every time.

Key Concepts

• Convergence: aₙ → L means for every ε > 0, all sufficiently late terms lie within ε of L.
• Divergence to +∞: terms eventually exceed any bound M.
• Oscillation: terms jump between different cluster values, e.g. (−1)ⁿ.
• Uniqueness of limit: a sequence cannot converge to two different numbers.
• Subsequence test: if two subsequences converge to different limits, the sequence diverges.

Formulas

• Convergence: ∀ε > 0 ∃N ∀n > N: |aₙ − L| < ε.
• Divergence to infinity: ∀M ∃N ∀n > N: aₙ > M.
• Algebra of limits: if aₙ → A and bₙ → B, then aₙ + bₙ → A + B and aₙbₙ → AB.
• Bounded + one cluster point: a bounded sequence with a single subsequential limit converges to it.

Worked Example

Consider aₙ = (−1)ⁿ + 1/n. Compute terms: n = 1 gives −1 + 1 = 0; n = 2 gives 1 + 0.5 = 1.5; n = 3 gives −1 + 1/3 ≈ −0.6667; n = 4 gives 1 + 0.25 = 1.25; n = 5 gives −1 + 0.2 = −0.8. The odd terms are −1 + 1/n → −1 and the even terms are 1 + 1/n → 1. Two subsequences with different limits means the sequence cannot converge — it diverges by oscillation, forever jumping between neighbourhoods of −1 and 1. Contrast with bₙ = 1 + 1/n: n = 10 gives 1.1, n = 1000 gives 1.001, and |bₙ − 1| = 1/n < ε whenever n > 1/ε, so bₙ converges to 1. Oscillation and blow-up are both divergence, but they look completely different.

Common Misconceptions

• Divergent means going to infinity. Oscillation, as in (−1)ⁿ, is also divergence — the sequence simply fails to settle.
• Bounded implies convergent. False — (−1)ⁿ is bounded between −1 and 1 yet diverges.
• Terms approaching each other implies convergence. The harmonic partial sums grow ever more slowly yet diverge to infinity.

Real-World Applications

• Series convergence tests decide whether infinite sums in physics are meaningful.
• Iterative algorithms are trusted only when their iterates converge.
• Signal processing distinguishes converging filters from oscillating ones.

Convergence means settling on one value; everything else is divergence. The subsequence test is the quickest way to prove a sequence does not converge.