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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 bn0, is called an alternating series
alternating series test
for an alternating series of either form, if bn+1bn for all integers n1 and bn0, 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 anM for all positive integers n
bounded below
a sequence {an} is bounded below if there exists a constant M such that Man 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 0anbn for all nN and n=1bn converges, then n=1an converges; if anbn0 for all nN 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 limnan0, 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=1arn1=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=1anand1f(x)dx

either both converge or both diverge
limit comparison test
suppose an,bn0 for all n1. If limnan/bnL0, then n=1an and n=1bn both converge or both diverge; if limnan/bn0 and n=1bn converges, then n=1an converges. If limnan/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=kn=1an=a1+a2+a3++ak
ratio test
for a series n=1an with nonzero terms, let ρ=limn|an+1/an|; if 0ρ<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=1anNn=1an satisfies the following estimate:
N+1f(x)dx<RN<Nf(x)dx
root test
for a series n=1an, let ρ=limnn|an|; if 0ρ<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
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