Convergence Tests

An infinite series is the sum of infinitely many terms of a sequence, written as ∑aₙ from n=1 to ∞. The central question is whether this endless addition produces a finite result (convergence) or grows without bound (divergence). Convergence tests are systematic tools that let us answer this question without having to compute the actual sum — which is often impossible in closed form. Four fundamental tests cover the vast majority of series encountered at high school and early university level: the nth-Term Test (also called the Divergence Test), the Comparison Test, the Ratio Test, and the Alternating Series Test. Each test has conditions under which it applies and a clear verdict it can deliver. No single test works on every series, so mathematicians learn to recognise which tool fits the structure of each problem. Mastering convergence tests builds critical analytical skills: you learn to estimate the size of terms, compare unknown series to known benchmarks like geometric series or p-series, and reason carefully about limits. These ideas underpin calculus, differential equations, physics simulations, and even the algorithms powering modern computers.

Key takeaways
  • nth-Term Test (Divergence Test)
  • Direct Comparison Test
  • Ratio Test
  • Alternating Series Test (Leibniz Test)

nth-Term Test (Divergence Test)

The simplest check: if the terms aₙ do not approach zero, the series MUST diverge. Formally, if lim(n→∞) aₙ ≠ 0, then ∑aₙ diverges. Example: ∑(n/(n+1)) diverges because aₙ → 1 ≠ 0. IMPORTANT CAVEAT: if lim aₙ = 0, the test is inconclusive — the series may still diverge (e.g., the harmonic series ∑1/n diverges even though 1/n → 0). This test can only confirm divergence, never convergence.

Direct Comparison Test

Compare an unfamiliar series to a known benchmark. If 0 ≤ aₙ ≤ bₙ for all n beyond some index N, then: (i) if ∑bₙ converges, then ∑aₙ converges (smaller than something finite ⟹ finite); (ii) if ∑aₙ diverges, then ∑bₙ diverges (bigger than something infinite ⟹ infinite). Common benchmarks: geometric series ∑rⁿ (converges for |r|<1), p-series ∑1/nᵖ (converges for p>1, diverges for p≤1). Example: ∑1/(n²+3) ≤ ∑1/n² and since ∑1/n² converges (p=2>1), so does ∑1/(n²+3).

Ratio Test

Best for series involving factorials or exponentials. Compute L = lim(n→∞) |a_{n+1}/aₙ|. Decision: if L < 1 the series converges absolutely; if L > 1 (or L = ∞) the series diverges; if L = 1 the test is inconclusive. The intuition is that when L < 1, consecutive terms shrink by a nearly constant ratio, like a geometric series with common ratio L. Example: for ∑n!/3ⁿ... wait, this diverges — compute |a_{n+1}/aₙ| = (n+1)/3 → ∞ > 1, so it diverges. For ∑3ⁿ/n!, the ratio is 3/(n+1) → 0 < 1, so it converges.

Alternating Series Test (Leibniz Test)

Applies to series of the form ∑(−1)ⁿbₙ or ∑(−1)^(n+1)bₙ where bₙ > 0. The series converges if: (i) the terms are eventually decreasing: bₙ₊₁ ≤ bₙ for all large n, AND (ii) the terms approach zero: lim(n→∞) bₙ = 0. Example: the alternating harmonic series 1 − 1/2 + 1/3 − 1/4 + ⋯ = ∑(−1)^(n+1)/n converges (to ln 2 ≈ 0.693) even though the ordinary harmonic series diverges. Bonus: the error in stopping at N terms is at most b_{N+1}.

The Key Equation

L=lim⁡n→∞∣an+1an∣⇒{L<1converges absolutelyL>1divergesL=1inconclusiveL = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| \quad \Rightarrow \quad \begin{cases} L < 1 & \text{converges absolutely} \\ L > 1 & \text{diverges} \\ L = 1 & \text{inconclusive} \end{cases}
  • L — Limiting ratio (dimensionless): The limit of |a_{n+1}/aₙ| as n → ∞; determines convergence behaviour
  • a_n — nth term (dimensionless): The general term of the series being tested
  • a_{n+1} — (n+1)th term (dimensionless): The term immediately following aₙ in the series

Worked Example

Problem. Determine whether the series ∑(n=1 to ∞) of 2ⁿ/n! converges or diverges.

∑n=1∞2nn!=e2−1≈6.389\sum_{n=1}^{\infty} \frac{2^n}{n!} = e^2 - 1 \approx 6.389

Answer. The series converges (by the Ratio Test, L = 0 < 1). Its sum equals e² − 1.

Figure
Diagram
Partial Sums: Alternating Harmonic vs Harmonic SeriesnSₙln2≈0.691234567Alternating harmonic (converges to ln 2)Harmonic series (diverges to ∞)
Origins · 1 minAugustin-Louis Cauchy and how this was discovered
AC
Augustin-Louis Cauchy
1789–1857 · French

Cauchy placed the theory of infinite series on rigorous foundations in his landmark 1821 textbook Cours d'analyse. He gave the first precise definition of convergence, developed the Ratio Test (sometimes called Cauchy's Ratio Test), and introduced the related Root Test. His insistence on rigour transformed informal 18th-century manipulations of series into a logically sound branch of mathematics. Cauchy was extraordinarily prolific — he published 789 papers in his lifetime, second only to Euler in mathematical output. Colleagues joked that the Paris Academy had to impose a page limit on papers specifically because of Cauchy's flood of submissions.

  • Geometric Series — The geometric series ∑rⁿ is the primary benchmark used in the Comparison Test and motivates the Ratio Test — a series 'behaves like' a geometric series when the ratio of consecutive terms approaches a constant L.
  • Partial Sums and Sequences — Convergence of a series is formally defined via convergence of its sequence of partial sums Sₙ — understanding sequences is prerequisite to understanding any convergence test.
  • Limits at Infinity — Every convergence test requires computing lim(n→∞) of some expression; fluency with limits at infinity is essential for applying the nth-Term, Ratio, and Alternating Series tests.
  • p-Series and the Harmonic Series — The p-series ∑1/nᵖ is the second key benchmark for the Comparison Test; knowing exactly when it converges (p > 1) or diverges (p ≤ 1) is indispensable for applying direct comparison.