Convergence and Divergence

An infinite series is the sum of infinitely many terms drawn from a sequence. At first glance, adding infinitely many numbers seems impossible — surely the total must grow without bound? Yet mathematics reveals a surprising truth: some infinite sums settle down to a single, finite value. When this happens, we say the series converges. When the partial sums instead grow without limit (or oscillate forever without settling), the series diverges. The key idea is the sequence of partial sums. Rather than trying to add all infinitely many terms at once, we watch what happens as we include more and more terms. If those running totals approach a definite limit L, the series converges to L. This framework, developed rigorously in the 18th and 19th centuries, underpins calculus, physics, engineering, and computer science — from computing the value of π to modelling electrical circuits. Understanding convergence and divergence requires careful thinking about limits. A series can diverge in different ways: its partial sums might shoot off to infinity, oscillate between two values, or behave erratically. Recognising these behaviours — and having reliable tests to distinguish them — is one of the central skills of this unit.

Key takeaways
  • From Sequence to Series
  • Partial Sums
  • The Definition of Convergence
  • The Divergence Test (nth-Term Test)
  • Geometric Series — The Key Example
  • Absolute vs Conditional Convergence

From Sequence to Series

Start with a sequence of terms a₁, a₂, a₃, … . The corresponding infinite series is the expression a₁ + a₂ + a₃ + … , written in sigma notation as Σ(n=1 to ∞) aₙ. This notation is shorthand — we cannot literally perform infinitely many additions, so we need a limiting process.

Partial Sums

Define the Nth partial sum Sₙ = a₁ + a₂ + … + aₙ. This is a finite, computable quantity. The sequence S₁, S₂, S₃, … tells us how the running total grows. For the series Σ(1/2ⁿ), we get S₁ = 1/2, S₂ = 3/4, S₃ = 7/8, S₄ = 15/16, and so on — these partial sums are clearly heading toward 1.

The Definition of Convergence

The series Σ aₙ converges to the sum L if and only if lim(N→∞) Sₙ = L, meaning that for every ε > 0, there exists N such that |Sₙ − L| < ε for all n > N. If no such L exists — because Sₙ → ∞, Sₙ → −∞, or Sₙ oscillates — the series diverges.

The Divergence Test (nth-Term Test)

A crucial first check: if lim(n→∞) aₙ ≠ 0, then Σ aₙ must diverge. The terms must shrink to zero for convergence to even be possible. However, the converse is false — terms going to zero does NOT guarantee convergence. The harmonic series Σ(1/n) has terms → 0, yet it diverges (its partial sums grow without bound, just very slowly).

Geometric Series — The Key Example

The geometric series Σ(n=0 to ∞) arⁿ = a + ar + ar² + … converges to a/(1−r) when |r| < 1, and diverges when |r| ≥ 1. This is one of the few series types with a clean closed-form sum. For example, Σ(n=0 to ∞) (1/3)ⁿ = 1/(1 − 1/3) = 3/2. This result is provable by computing Sₙ = a(1 − rⁿ)/(1 − r) and taking the limit as n → ∞.

Absolute vs Conditional Convergence

A series Σ aₙ converges absolutely if Σ |aₙ| converges. Absolute convergence implies convergence. A series converges conditionally if Σ aₙ converges but Σ |aₙ| diverges. The alternating harmonic series Σ(−1)ⁿ⁺¹/n = 1 − 1/2 + 1/3 − 1/4 + … converges conditionally to ln 2, even though Σ 1/n diverges.

The Key Equation

S=∑n=0∞arn=a1−r,∣r∣<1S = \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}, \quad |r| < 1
  • S — Sum (dimensionless): The limit of the partial sums; the value the series converges to.
  • a — First term (dimensionless): The initial term of the series (the n = 0 term).
  • r — Common ratio (dimensionless): The factor by which each term is multiplied to get the next. Must satisfy |r| < 1 for convergence.
  • n — Index (dimensionless): The summation index running from 0 to infinity.

Worked Example

Problem. Does the series Σ(n=1 to ∞) 4·(2/3)ⁿ converge or diverge? If it converges, find its sum.

∑n=1∞4(23)n=8\sum_{n=1}^{\infty} 4\left(\frac{2}{3}\right)^n = 8

Answer. The series converges to 8.

Figure
Diagram
Partial Sums of a Convergent Geometric SeriesΣ (1/2)ⁿ, n=1 to ∞ → converges to 100.50.750.8750.93751.0L=1n12345S₁=0.5S₂=0.75S₃=0.875S₄→1Partial sums Sₙ approach the limit L = 1 (red dashed line)
Origins · 1 minAugustin-Louis Cauchy and how this was discovered
AC
Augustin-Louis Cauchy
1789–1857 · French

Cauchy was the first mathematician to give a rigorous definition of the convergence of an infinite series, introducing the concept of the limit of partial sums in his landmark 1821 textbook Cours d'analyse. He also proved the Cauchy convergence criterion: a series converges if and only if its partial sums form a Cauchy sequence, where |Sₙ − Sₘ| can be made arbitrarily small for all sufficiently large m and n. Cauchy was extraordinarily prolific — he published around 800 research papers, second only to Euler in mathematical output. He reportedly worked so fast that the Paris Academy of Sciences had to create a rule limiting members to one submission per month specifically because of him.

  • Geometric Sequences and Series — The geometric series is the primary closed-form example of a convergent infinite series; mastering it is essential before studying general convergence tests.
  • Limits of Sequences — Convergence of a series is defined via the limit of its partial sums, so a solid understanding of sequence limits is a prerequisite.
  • Convergence Tests (Ratio, Root, Comparison) — Once the definition of convergence is understood, these systematic tests let you determine convergence for a wide variety of series.
  • The Harmonic Series — The harmonic series Σ(1/n) is the classic example showing that terms approaching zero does not guarantee convergence — a crucial conceptual counterexample.