Series Tests
1. Limit Divergence Test:
If l im a n is not zero then the series
n! 1
X
1
a n diverges
n=0
This is the first test you should check! Most of the time you can check in your head if the limit is
not going to be zero.
2. Integral Test:
If a n is decreasing and bounded above, then
X
1
a n converges if and only if
n=0
converges.
Z
1
a xdx
0
Note: This test is used to prove p-series test (i) and (ii) below.
3. P-Series:
(i) The series
(ii) The series
1
X
n=0
1
X
1 converges for p > 1 and diverges for p ∙ 1 .
np
1
p converges for p > 1 and diverges for p ∙ 1 .
n
(
ln
n
)
n= 2
4. Geometric Series:
The series
1
X
n=0
c r n converges for r < 1 and diverges for p ¸ 1.
If r < 1, then the series converges to the value
c .
1¡r
5. Comparison Test:
If
1
X
an <
n=0
If
1
X
n=0
1
X
b n and
n=0
an >
1
X
n=0
1
X
b n converges, then
n=0
b n and
1
X
n=0
b n diverges, then
1
X
a n converges too.
n=0
1
X
a n diverges too.
n=0
Remember: If you are short, and you want to prove you are short, find someone taller than
yourself who is considered to be short. On the other hand, If you are tall and want to prove that
you are tall, find someone shorter than yourself that is considered to be tall.
6. Limit Comparison Test:
a
If l im n = c where c is a nonzero constant, then
n! 1 b
n
1
X
n=0
an »
1
X
b n and they either both
n= 0
converge or both diverge.
Remember: If the limit of the ratios of the two sequences is a number that isn’t zero (or infinity)
then what we are saying is that the two sequences are very similar, and so their series should
behave in the same way - meaning they should either both converge or both diverge.
The following concept precedes both the ratio and the root tests.
Remember that
X
1
an =
n=0
X
N
an +
n=0
X
1
a n for some large number N. Since the first
n= N
sum is finite, it always converges, therefore, we say that a series
1
X
a n converges if its tail
n=0
converges (the tail being the second term in the sum). The idea is that every series “eventually”
starts looking geometric. Meaning that every series has a tail that looks geometric. Formally
speaking:
1
X
an »
n=0
Since
1
X
N
X
1
X
n=0
c rn
n=N
a n converges if
n= 0
n=N
an +
1
X
1
X
n=N
a n converges, then we can say that
1
X
a n converges if
n=0
c r n converges. But we happen to know that a geometric series converges if j r j < 1
and diverges if j r j > 1 (for now, we will not concern ourselves with what happens exactly at
1 because we are dealing with approximations.)
So the question reduces to finding out what r is! How do we do that? Well there are two ways:
. We could find the ratio of two consecutive terms in the tail of a series.
. We could take the n th root of a n .
The first method is what we call the ratio test, and the second method is the root test. They are
both methods to find r. The conclusions drawn regarding the convergence of the series is the
same for both tests.
7. Ratio Test:
Find l im
n! 1
a
j an + 1 j = j r j .
n
If j r j < 1, then
If j r j > 1, then
1
X
n=0
1
X
a n converges.
a n diverges.
n=0
If j r j = 1 , then the test fails.
8. Root Test:
Find l im
1
n
j (an ) j = j r j .
n! 1
1
X
If j r j < 1, then
a n converges.
If j r j > 1, then
n=0
1
X
a n diverges.
n=0
If j r j = 1 , then the test fails.