Math · Calculus II · Concept

Infinite series and convergence tests

An infinite series adds infinitely many terms; it converges when its partial sums approach a finite limit. A geometric series converges exactly when |r| < 1. Terms that do not shrink to zero force divergence, and the integral, comparison, ratio and alternating series tests settle most of the series a course meets.

Partial sums

The partial sum S_N adds the first N terms. The series converges to S if S_N approaches S as N grows; otherwise it diverges. A table of partial sums can suggest a limit, but it cannot prove one.

S_{N} = \sum_{n = 1}^{N} a_{n}

Geometric series

A geometric series multiplies each term by the same ratio r. It converges exactly when |r| < 1, and then its sum is the first term divided by 1 − r.

\sum_{n = 1}^{\infty} a r^{n - 1} = \frac{a}{1 - r}, ∣ r ∣ < 1

The divergence test

If the terms do not approach 0, the series diverges. The converse is false: the harmonic series 1 + 1/2 + 1/3 + ⋯ has terms approaching 0 and still diverges.

The integral test and p-series

If aₙ = f(n) for a function f that is positive and decreasing, the series and the improper integral of f from 1 to ∞ converge or diverge together. For p-series this gives a clean rule.

\sum_{n = 1}^{\infty} \frac{1}{n^{p}} converges p > 1

Comparison tests

A series of positive terms that is smaller, term by term, than a convergent series converges; one that is larger than a divergent series diverges. The limit comparison test compares aₙ with a simpler bₙ through the limit of aₙ/bₙ.

The ratio test

If the ratio of consecutive terms approaches L < 1, the series converges absolutely; if L > 1, it diverges; if L = 1, the test decides nothing. It suits series with powers and factorials.

L = lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} |

Alternating series

If the terms alternate in sign, shrink in size and approach 0, the series converges, and stopping after N terms leaves an error no larger than the first term left out. A series that converges only because of its alternating signs converges conditionally.

Common mistakes

Concluding convergence because the terms approach 0: the harmonic series is the counterexample.
Using the geometric sum a/(1 − r) when |r| ≥ 1.
Treating L = 1 in the ratio test as a verdict: it decides nothing, as for every p-series.
Starting the geometric sum at the wrong term: a must be the first term actually added.

Key terms

Infinite series: A sum of infinitely many terms, a₁ + a₂ + a₃ + ⋯. Its sum is the limit of its partial sums, if that limit exists.
Partial sum: The sum of the first n terms of a series, Sₙ = a₁ + ⋯ + aₙ. If the partial sums approach a limit, that limit is the series’ sum.
Series convergence: A series converges when its partial sums approach a finite number. Terms going to zero is necessary but not enough: the harmonic series 1 + 1/2 + 1/3 + ⋯ diverges.
Geometric series: A series in which each term is the previous one times a fixed ratio r: a + ar + ar² + ⋯. It converges to a/(1 − r) when |r| < 1 and diverges otherwise, unless a = 0.
Divergence test: If the terms of a series don’t approach zero, the series diverges. If they do approach zero, this test tells you nothing.
Harmonic series: The series 1 + 1/2 + 1/3 + ⋯. Its terms go to zero, yet its partial sums grow without bound, so it diverges.
Integral test: If aₙ = f(n) for a positive, continuous, decreasing function f, then Σaₙ and ∫₁^∞ f(x) dx either both converge or both diverge.
p-series: The series Σ 1/nᵖ. It converges when p > 1 and diverges when p ≤ 1, as the harmonic series (p = 1) shows.
Comparison test: A test for series of positive terms: a series smaller, term by term, than a convergent series converges, and one larger than a divergent series diverges. The limit comparison test compares two series through the limit of the ratio of their terms.
Ratio test: Find L = lim |aₙ₊₁/aₙ|. If L < 1 the series converges absolutely, if L > 1 it diverges, and if L = 1 the test can’t decide.
Alternating series: A series whose terms switch sign, such as 1 − 1/2 + 1/3 − ⋯. If the sizes of the terms decrease to zero it converges, and stopping early is off by at most the next term’s size.
Conditional convergence: A series that converges only because its positive and negative terms cancel: with every term made positive, it diverges. The alternating harmonic series is an example, and reordering its terms can change its sum.

Work through an example

Find the sum of 3 + 3/2 + 3/4 + 3/8 + ⋯.

Sum a geometric series →

Test a p-series with the integral test →

Test a series with the ratio test →

Test an alternating series →

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Sum it in Sequences & infinite series Check the sum in Math Open worked example on a board Infinite geometric sum in Math Reference

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