Calculus Volume 2

# Key Terms

absolute convergence
if the series $∑n=1∞|an|∑n=1∞|an|$ converges, the series $∑n=1∞an∑n=1∞an$ is said to converge absolutely
alternating series
a series of the form $∑n=1∞(−1)n+1bn∑n=1∞(−1)n+1bn$ or $∑n=1∞(−1)nbn,∑n=1∞(−1)nbn,$ where $bn≥0,bn≥0,$ is called an alternating series
alternating series test
for an alternating series of either form, if $bn+1≤bnbn+1≤bn$ for all integers $n≥1n≥1$ and $bn→0,bn→0,$ 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 ${an}{an}$ is bounded above if there exists a constant $MM$ such that $an≤Man≤M$ for all positive integers $nn$
bounded below
a sequence ${an}{an}$ is bounded below if there exists a constant $MM$ such that $M≤anM≤an$ for all positive integers $nn$
bounded sequence
a sequence ${an}{an}$ is bounded if there exists a constant $MM$ such that $|an|≤M|an|≤M$ for all positive integers $nn$
comparison test
if $0≤an≤bn0≤an≤bn$ for all $n≥Nn≥N$ and $∑n=1∞bn∑n=1∞bn$ converges, then $∑n=1∞an∑n=1∞an$ converges; if $an≥bn≥0an≥bn≥0$ for all $n≥Nn≥N$ and $∑n=1∞bn∑n=1∞bn$ diverges, then $∑n=1∞an∑n=1∞an$ diverges
conditional convergence
if the series $∑n=1∞an∑n=1∞an$ converges, but the series $∑n=1∞|an|∑n=1∞|an|$ diverges, the series $∑n=1∞an∑n=1∞an$ 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 ${an}{an}$ for which there exists a real number $LL$ such that $anan$ is arbitrarily close to $LL$ as long as $nn$ is sufficiently large
divergence of a series
a series diverges if the sequence of partial sums for that series diverges
divergence test
if $limn→∞an≠0,limn→∞an≠0,$ then the series $∑n=1∞an∑n=1∞an$ diverges
divergent sequence
a sequence that is not convergent is divergent
explicit formula
a sequence may be defined by an explicit formula such that $an=f(n)an=f(n)$
geometric sequence
a sequence ${an}{an}$ in which the ratio $an+1/anan+1/an$ is the same for all positive integers $nn$ is called a geometric sequence
geometric series
a geometric series is a series that can be written in the form
$∑n=1∞arn−1=a+ar+ar2+ar3+⋯∑n=1∞arn−1=a+ar+ar2+ar3+⋯$
harmonic series
the harmonic series takes the form
$∑n=1∞1n=1+12+13+⋯∑n=1∞1n=1+12+13+⋯$
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
$a1+a2+a3+⋯=∑n=1∞ana1+a2+a3+⋯=∑n=1∞an$
integral test
for a series $∑n=1∞an∑n=1∞an$ with positive terms $an,an,$ if there exists a continuous, decreasing function $ff$ such that $f(n)=anf(n)=an$ for all positive integers $n,n,$ then
$∑n=1∞anand∫1∞f(x)dx∑n=1∞anand∫1∞f(x)dx$

either both converge or both diverge
limit comparison test
suppose $an,bn≥0an,bn≥0$ for all $n≥1.n≥1.$ If $limn→∞an/bn→L≠0,limn→∞an/bn→L≠0,$ then $∑n=1∞an∑n=1∞an$ and $∑n=1∞bn∑n=1∞bn$ both converge or both diverge; if $limn→∞an/bn→0limn→∞an/bn→0$ and $∑n=1∞bn∑n=1∞bn$ converges, then $∑n=1∞an∑n=1∞an$ converges. If $limn→∞an/bn→∞,limn→∞an/bn→∞,$ and $∑n=1∞bn∑n=1∞bn$ diverges, then $∑n=1∞an∑n=1∞an$ diverges
limit of a sequence
the real number $LL$ 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 $∑n=1∞1/np∑n=1∞1/np$
partial sum
the $kthkth$ partial sum of the infinite series $∑n=1∞an∑n=1∞an$ is the finite sum
$Sk=∑n=1kan=a1+a2+a3+⋯+akSk=∑n=1kan=a1+a2+a3+⋯+ak$
ratio test
for a series $∑n=1∞an∑n=1∞an$ with nonzero terms, let $ρ=limn→∞|an+1/an|;ρ=limn→∞|an+1/an|;$ if $0≤ρ<1,0≤ρ<1,$ the series converges absolutely; if $ρ>1,ρ>1,$ the series diverges; if $ρ=1,ρ=1,$ the test is inconclusive
recurrence relation
a recurrence relation is a relationship in which a term $anan$ in a sequence is defined in terms of earlier terms in the sequence
remainder estimate
for a series $∑n=1∞an∑n=1∞an$ with positive terms $anan$ and a continuous, decreasing function $ff$ such that $f(n)=anf(n)=an$ for all positive integers $n,n,$ the remainder $RN=∑n=1∞an−∑n=1NanRN=∑n=1∞an−∑n=1Nan$ satisfies the following estimate:
$∫N+1∞f(x)dx
root test
for a series $∑n=1∞an,∑n=1∞an,$ let $ρ=limn→∞|an|n;ρ=limn→∞|an|n;$ if $0≤ρ<1,0≤ρ<1,$ the series converges absolutely; if $ρ>1,ρ>1,$ the series diverges; if $ρ=1,ρ=1,$ the test is inconclusive
sequence
an ordered list of numbers of the form $a1,a2,a3,…a1,a2,a3,…$ 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 $anan$ in the sequence ${an}{an}$ is called the $nthnth$ term of the sequence
unbounded sequence
a sequence that is not bounded is called unbounded