Math 166, Chapter 9 Tests for Convergence or Divergence
Nth Term Test for Divergence
 If lim an  0 or lim an does not exist, then the series diverges.
n 
n 
Useful Series
 Geometric Series

o
 ar
n 1
a 0
=(a+ar+ar2+ar3+…)
n 1

o
If r <1 then the series converges, otherwise the series diverges
o
S
a
if the series converges
1 r
Harmonic Series

o
1
n
the series diverges
n 1

Telescoping Series

1
1
 n  n 1
o
General example
o
Write out terms and see what cancels. Then write the equation for Sn and find lim S n .
n 1
n 
P-series Test


Given the series
1
n
n 1
o
o
p
If p>1, the series converges
If p  1, the series diverges
Integral Test
 Let an=f(x) and f(x) be continuous, positive, and non-increasing on the interval [N,  )


o
a
n 1
and
n
 f ( x)dx converge or diverge together
N
Ordinary Comparison Test
 Let 0  an  bn for n  N
o If
bn converges then so does
o

If  a
n
a
diverges then so does  b
n
n
Limit Comparison Test

Suppose that an  0 , bn  0 and lim
n 
o
If 0<L<  , then
o
If L=0 and
b
n
a
n
and
an
L
bn
b
n
converges, then
converge or diverge together
a
n
converges
Ratio Test

Let
a
n
o
o
o
be a series of positive terms and lim
n 
If   1 , the series converges
If   1 or if    , the series diverges
If   1 , the test is inconclusive
an 1

an
Alternating Series Test
 Let a1  a2  a3  a4   be an alternating series with an  an 1  0 .
o
If lim an  0 then the series converges.
o
The error made by using the sum Sn of the first n terms to approximate the sum S of the
series is not more than an 1 .
n 
Absolute Convergence Test
 If
un converges, then

u
n
converges.
Absolute Ratio Test

Let
u
o
o
o
n
be a series of nonzero terms and suppose that lim
n 
un 1
un

If   1 , the series converges absolutely (hence converges).
If   1 , the series diverges.
If   1 , the test is inconclusive.