SUMMARY – INFINITE SERIES
MA 182
nth-TERM TEST

If lim an  0 , then
a
If lim an  0 , then
a
n 
n 
n
diverges.
n
may converge or it may diverge.
n 1

n 1
GEOMETRIC SERIES

The series
 ar
n 1
 a  ar  ar 2    ar n 1   converges if r  1 and diverges if r  1 .
n 1
p-SERIES

The series
1
n
n 1
p
converges if p>1 and diverges if p  1 .
INTEGRAL TEST
If f is continuous, positive, and decreasing on [1, ] and if an  f (n) , then


a
n 1
n
converges if
 f ( x)dx
converges, and
1

 an diverges if
n 1

 f ( x)dx
diverges.
1
COMPARISON TESTS

Suppose that
 an and
n 1

 bn are two series of positive terms and the convergence of
n 1

b
n 1
n
is
known.
Basic Comparison Test
(1)
If there exists a number N such that an  bn for n > N, and

b
n 1
n
converges, then

a
n 1
(2)
n
converges.
If there exists a number N such that an  bn for n>N, and

 bn diverges, then
n 1

a
n 1
n
diverges.
Limit Comparison Test
an
 L , where L is a non-zero finite number, then
n  b
n
If lim

 an diverges if
n 1
(3)
If lim
n 
If lim
n
 an converges if
n 1

b
n 1
n
converges, and

b
n 1
diverges.
n
RATIO TEST
a
(1)
If lim n 1  L <1, the series
n  a
n
(2)


a
n 1
n
is absolutely convergent and therefore convergent.
an 1
a
 L >1 or if , lim n1   , the series
n  a
an
n

a
n 1
n
is divergent
an1
 1 , this test has no conclusion and another test must be used.
an
ALTERNATING SERIES TEST

If the alternating series
 (1)n1 bn or
n 1
(1)
lim bn  0
n 
and
(2)

 (1) b
n
n 1
n
satisfies
bn 1  bn for all n beyond some finite number N,
then the series is convergent.
ALTERNATING SERIES ESTIMATION THEOREM
If S is the sum of an alternating series that satisfies conditions (1) and (2) in the Alternating Series
Test, then the size of the error in using the first n terms to estimate S is less than the first terms to be
omitted, that is, Rn  S  Sn  bn1
NOTES:
(1)
The convergence or divergence of a series is not affected if each term is multiplied by a nonzero finite constant.
(2)
The sum of two convergent series converges. The sum of a convergent and a divergent
series diverges.
(3)
In general, a series “behaves like” the series which is found by considering the dominant
term in the numerator and the dominant term in the denominator.