Key Terms
- absolute convergence
- if the series converges, the series is said to converge absolutely
- alternating series
- a series of the form or where is called an alternating series
- alternating series test
- for an alternating series of either form, if for all integers and 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 is bounded above if there exists a constant such that for all positive integers
- bounded below
- a sequence is bounded below if there exists a constant such that for all positive integers
- bounded sequence
- a sequence is bounded if there exists a constant such that for all positive integers
- comparison test
- if for all and converges, then converges; if for all and diverges, then diverges
- conditional convergence
- if the series converges, but the series diverges, the series 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 for which there exists a real number such that is arbitrarily close to as long as is sufficiently large
- divergence of a series
- a series diverges if the sequence of partial sums for that series diverges
- divergence test
- if then the series diverges
- divergent sequence
- a sequence that is not convergent is divergent
- explicit formula
- a sequence may be defined by an explicit formula such that
- geometric sequence
- a sequence in which the ratio is the same for all positive integers is called a geometric sequence
- geometric series
- a geometric series is a series that can be written in the form
- harmonic series
- the harmonic series takes the form
- 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
- integral test
- for a series with positive terms if there exists a continuous, decreasing function such that for all positive integers then
either both converge or both diverge
- limit comparison test
- suppose for all If then and both converge or both diverge; if and converges, then converges. If and diverges, then diverges
- limit of a sequence
- the real number 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
- partial sum
- the partial sum of the infinite series is the finite sum
- ratio test
- for a series with nonzero terms, let if the series converges absolutely; if the series diverges; if the test is inconclusive
- recurrence relation
- a recurrence relation is a relationship in which a term in a sequence is defined in terms of earlier terms in the sequence
- remainder estimate
- for a series with positive terms and a continuous, decreasing function such that for all positive integers the remainder satisfies the following estimate:
- root test
- for a series let if the series converges absolutely; if the series diverges; if the test is inconclusive
- sequence
- an ordered list of numbers of the form 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 in the sequence is called the term of the sequence
- unbounded sequence
- a sequence that is not bounded is called unbounded