Key Terms
- absolute convergence
- if the series ∑∞n=1|an| converges, the series ∑∞n=1an is said to converge absolutely
- alternating series
- a series of the form ∑∞n=1(−1)n+1bn or ∑∞n=1(−1)nbn, where bn≥0, is called an alternating series
- alternating series test
- for an alternating series of either form, if bn+1≤bn for all integers n≥1 and bn→0, then an alternating series converges
- arithmetic sequence
- a sequence in which the difference between every pair of consecutive terms is the same is called an arithmetic sequence
- bounded above
- a sequence {an} is bounded above if there exists a constant M such that an≤M for all positive integers n
- bounded below
- a sequence {an} is bounded below if there exists a constant M such that M≤an for all positive integers n
- bounded sequence
- a sequence {an} is bounded if there exists a constant M such that |an|≤M for all positive integers n
- comparison test
- if 0≤an≤bn for all n≥N and ∑∞n=1bn converges, then ∑∞n=1an converges; if an≥bn≥0 for all n≥N and ∑∞n=1bn diverges, then ∑∞n=1an diverges
- conditional convergence
- if the series ∑∞n=1an converges, but the series ∑∞n=1|an| diverges, the series ∑∞n=1an is said to converge conditionally
- convergence of a series
- a series converges if the sequence of partial sums for that series converges
- convergent sequence
- a convergent sequence is a sequence {an} for which there exists a real number L such that an is arbitrarily close to L as long as n is sufficiently large
- divergence of a series
- a series diverges if the sequence of partial sums for that series diverges
- divergence test
- if limn→∞an≠0, then the series ∑∞n=1an diverges
- divergent sequence
- a sequence that is not convergent is divergent
- explicit formula
- a sequence may be defined by an explicit formula such that an=f(n)
- geometric sequence
- a sequence {an} in which the ratio an+1/an is the same for all positive integers n is called a geometric sequence
- geometric series
- a geometric series is a series that can be written in the form
∞∑n=1arn−1=a+ar+ar2+ar3+⋯
- harmonic series
- the harmonic series takes the form
∞∑n=11n=1+12+13+⋯
- index variable
- the subscript used to define the terms in a sequence is called the index
- infinite series
- an infinite series is an expression of the form
a1+a2+a3+⋯=∞∑n=1an
- integral test
- for a series ∑∞n=1an with positive terms an, if there exists a continuous, decreasing function f such that f(n)=an for all positive integers n, then
∞∑n=1anand∫∞1f(x)dx
either both converge or both diverge
- limit comparison test
- suppose an,bn≥0 for all n≥1. If limn→∞an/bn→L≠0, then ∑∞n=1an and ∑∞n=1bn both converge or both diverge; if limn→∞an/bn→0 and ∑∞n=1bn converges, then ∑∞n=1an converges. If limn→∞an/bn→∞, and ∑∞n=1bn diverges, then ∑∞n=1an diverges
- limit of a sequence
- the real number L to which a sequence converges is called the limit of the sequence
- monotone sequence
- an increasing or decreasing sequence
- p-series
- a series of the form ∑∞n=11/np
- partial sum
- the kth partial sum of the infinite series ∑∞n=1an is the finite sum
Sk=k∑n=1an=a1+a2+a3+⋯+ak
- ratio test
- for a series ∑∞n=1an with nonzero terms, let ρ=limn→∞|an+1/an|; if 0≤ρ<1, the series converges absolutely; if ρ>1, the series diverges; if ρ=1, the test is inconclusive
- recurrence relation
- a recurrence relation is a relationship in which a term an in a sequence is defined in terms of earlier terms in the sequence
- remainder estimate
- for a series ∑∞n=1an with positive terms an and a continuous, decreasing function f such that f(n)=an for all positive integers n, the remainder RN=∑∞n=1an−∑Nn=1an satisfies the following estimate:
∫∞N+1f(x)dx<RN<∫∞Nf(x)dx
- root test
- for a series ∑∞n=1an, let ρ=limn→∞n√|an|; if 0≤ρ<1, the series converges absolutely; if ρ>1, the series diverges; if ρ=1, the test is inconclusive
- sequence
- an ordered list of numbers of the form a1,a2,a3,… is a sequence
- telescoping series
- a telescoping series is one in which most of the terms cancel in each of the partial sums
- term
- the number an in the sequence {an} is called the nth term of the sequence
- unbounded sequence
- a sequence that is not bounded is called unbounded