An infinite series Σaₙ converges when its sequence of partial sums approaches a finite limit. The nth term test can only show divergence (if aₙ ⋮→ 0). Key families have quick rules: a geometric series converges iff |r| < 1, and a p-series Σ1/nᵖ iff p > 1. For everything else you choose among the integral, direct/limit comparison, alternating series, and ratio tests — the last being the go-to for factorials and powers.
ConvergenceGeometric & pRatio Test
Key Concept 2
Convergence can be absolute or conditional, and error is bounded
A series converges absolutely when Σ|aₙ| converges (which guarantees plain convergence); it converges conditionally when Σaₙ converges but Σ|aₙ| diverges, like the alternating harmonic series. When you approximate a sum with a partial sum, the alternating series error bound caps the error at the first omitted term, and the Lagrange error bound caps a Taylor approximation’s error using a bound on the next derivative.
AbsoluteConditionalError Bounds
Key Concept 3
Series represent functions: Taylor, Maclaurin, and power series
A Taylor polynomial Pₙ(x) = Σ f⁽ᵏ⁾(a)/k!·(x−a)ᵏ approximates f near x = a; a Maclaurin series is the special case a = 0. A power series converges on an interval whose radius of convergence R comes from the ratio test — with the endpoints tested separately. You should know the series for eˣ, sin x, cos x, and 1/(1−x), and generate new ones by differentiating, integrating, or substituting into them.
Taylor / MaclaurinPower SeriesKnown Series
Infinite series
Σaₙ, the sum of infinitely many terms.
Basics
Partial sum
Sₙ = a₁ + … + aₙ; convergence is its limit.
Basics
Geometric series
Σarⁿ; converges iff |r| < 1 to a/(1−r).
Series
p-series
Σ1/nᵖ; converges iff p > 1.
Series
Harmonic series
Σ1/n; the p = 1 case, which diverges.
Series
nth term test
aₙ ⋮→ 0 ⇒ the series diverges.
Tests
Integral test
Compares Σaₙ to ∫₁∞ f(x) dx.
Tests
Comparison / limit comparison
Compares to a known convergent/divergent series.
Tests
Alternating series test
Decreasing terms → 0 ⇒ convergence.
Tests
Ratio test
L = lim|aₙ₊₁/aₙ|; <1 converges, >1 diverges.
Tests
Absolute convergence
Σ|aₙ| converges (⇒ Σaₙ converges).
Convergence
Conditional convergence
Σaₙ converges but Σ|aₙ| diverges.
Convergence
Alternating error bound
|error| ≤ |first omitted term|.
Error
Lagrange error bound
|Rₙ| ≤ (M/(n+1)!)|x−a|ⁿ⁺¹.
Error
Taylor polynomial
Pₙ(x) = Σ f⁽ᵏ⁾(a)/k!·(x−a)ᵏ.
Series Rep
Maclaurin series
A Taylor series centered at a = 0.
Series Rep
Radius of convergence
R with convergence for |x−a| < R.
Power Series
Interval of convergence
All x for which the power series converges (test endpoints).