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Calculus II (Integration & Series) Unit 8: Sequences & convergence tests

An infinite series adds infinitely many terms. Convergence tests tell you whether the sum is a finite number.

Covers: Sequences, geometric, nth-term, integral, p-series, comparison, ratio, alternating series tests

Study guide

A sequence is an ordered list a₁, a₂, …; it converges if its terms approach a finite limit. A series adds the terms of a sequence: Σ aₙ. Its partial sums Sₙ form a sequence, and the series converges if those partial sums approach a finite value.

Several tests decide convergence. The nth-term test: if lim aₙ ≠ 0, the series diverges (but if the limit is 0, the test tells you nothing). Geometric series Σ arⁿ converge to a/(1 − r) when |r| 1; the harmonic series Σ 1/n diverges.

The integral test compares a positive, decreasing series to the integral of the matching function. The direct comparison and limit comparison tests compare a series to a known one, usually a p-series or geometric series. The ratio test computes L = lim |aₙ₊₁/aₙ|: the series converges absolutely if L 1, and the test is inconclusive if L = 1. It works best with factorials and exponentials.

The alternating series test says Σ (−1)ⁿbₙ converges if bₙ decreases to 0, and the error from stopping is at most the next term. A series converges absolutely if Σ |aₙ| converges, and conditionally if it converges but Σ |aₙ| doesn't.

When you use a test, state its conditions. The alternating series test needs terms that are positive, decreasing and approaching 0. The integral test needs f positive, continuous and decreasing. Limit comparison: if aₙ > 0, bₙ > 0 and lim aₙ/bₙ = c with 0 < c < ∞, then Σaₙ and Σbₙ both converge or both diverge. Compare with the dominant terms, such as 1/n² for (n + 1)/(n³ + 2).

To find the limit of a sequence, treat n as a continuous variable and use limit techniques, including L'Hospital's rule. Useful facts: n!/nⁿ → 0 and (1 + 1/n)ⁿ → e. A geometric series sums to (first term)/(1 − r), where the first term is whatever the first term actually is. For example, Σₙ₌₂^∞ (1/3)ⁿ = (1/9)/(1 − 1/3) = 1/6.

Key ideas

• nth-term test: if aₙ doesn't go to 0, the series diverges. (If it does, the test says nothing.)
• Geometric series Σ arⁿ converge when |r| < 1.
• p-series Σ 1/nᵖ converge when p > 1.
• Comparison and limit comparison: compare to a series you already know.
• Ratio test: lim |aₙ₊₁/aₙ| 1 diverges, = 1 inconclusive. Great for factorials and exponentials.
• Alternating series test: terms decrease to 0 ⇒ converges. Absolute vs conditional convergence.

Formulas and rules

Common mistakes

• Using the nth-term test to prove convergence.
• Forgetting the harmonic series Σ 1/n diverges.

Words to know

nth-term test

If lim aₙ ≠ 0, the series diverges.

Harmonic series

Σ 1/n; it diverges.

P-series

Σ 1/nᵖ; converges when p > 1.

Ratio test (L = 1)

Inconclusive; try another test.

Ratio test

L = lim |aₙ₊₁/aₙ|: L 1 diverges. Good for factorials and exponentials.

Conditional convergence

The series converges but the series of absolute values diverges.

Geometric series

Σ arⁿ converges to a/(1 − r) when |r| < 1.

Sequence

An ordered list a₁, a₂, …; converges if lim aₙ is finite.

Series

Σ aₙ: the limit of the partial sums.

Partial sum

Sₙ = a₁ + a₂ + … + aₙ

Integral test

For positive decreasing aₙ = f(n), Σ aₙ and ∫ f(x) dx both converge or both diverge.

Direct comparison test

Compare to a known series: smaller than convergent converges; bigger than divergent diverges.

Limit comparison test

If lim aₙ/bₙ is positive and finite, both series do the same thing.

Alternating series test

Σ (−1)ⁿbₙ converges if bₙ decreases to 0.

Absolute convergence

Σ |aₙ| converges.

Sample practice questions

29 more questions like these, from level 1 to 5, are in the practice set.