How Series Convergence Tests Work
An infinite series Σa_n converges if its partial sums settle toward a finite number — otherwise it diverges. No single test handles every series, so mathematicians built a toolbox. The strategy: try the cheapest test first and escalate only when it comes back inconclusive.
The Test Ladder
- nth-term test: if a_n does not → 0, the series diverges. (Passing it proves nothing.)
- Geometric: Σar^(n−1) converges iff |r| < 1, summing to a/(1−r).
- p-series: Σ1/n^p converges iff p > 1 — the harmonic series (p = 1) famously diverges.
- Ratio test: if |a_(n+1)/a_n| → L < 1, converges; L > 1, diverges; L = 1, inconclusive.
- Root test: same logic with |a_n|^(1/n) — best for terms with nth powers.
Series are the engine behind Taylor expansions — see our Taylor series calculator.