Series Tests
 0 then the series  a

1. The test for divergence: If lim an
n
2. The integral test: If


1
n 1
n
diverges. Always use this test first.

ax dx  L then the series  an converges. If


1
n 1
ax dx  L then the series

a
n 1
n
diverges. Use this if it is easy to integrate an .
3. The p-series test. If an 
k
use this test. If p  1 the series
np

a
n 1
n
diverges. If p  1 the series

a
n 1
n
converges.

4. The comparison test. If
b
n 1
n

is a known convergent series and
n 1

converges. If

b
n 1
is a known divergent series and
n
a
a
n 1
n
n

<
b
n 1
n
then the series

>
b
n 1
n
then the series diverges. If

a
n 1
n
can be compared to a series that is known to converge or diverge then use this test.
bn
 C where C is a finite number and C > 0 then both series
n a
n
5. Limit comparison test. If lim

converge or diverge. Use this if you can compare
a
n 1
n
to a series but you can’t tell if the
series is greater or smaller than the compared series.

6. Alternating series test. If
  1
n 1
n 1
an (an alternating series) where an1  an and lim an  0
n
then the series converges. Use this for alternating series.
a
a
7. Ratio Test. If lim n 1  1 the series is absolutely convergent. If lim n 1  1 the series is
n  a
n  a
n
n
an1
 1 then you need more tests. Use this test if the others aren’t obvious or
n  a
n
if an contains a mixture of factorials, geometric, and p-series.
divergent. If lim
8. n root test : If lim n an  L  1 then the series is absolutely convergent. If
th
n
lim n an  L  1 or lim n an   then the series diverges. Use this if your general term is in the
n
n
form of  an  .
n