THE INTEGRAL TEST: Suppose is a continuous, positive, decreasing function on and let
Then the series is convergent if and only if the improper integral
is convergent. In other words:

(a) If is convergent, then is convergent.

(b) If is divergent, then is divergent.

REMARK: Don't use the Integral Test to evaluate series, because in general

EXAMPLES:

1. is divergent, because is continuous, positive, decreasing and
is divergent by the -test for improper integrals, since

2. is divergent, because is continuous, positive, decreasing and
is divergent by the -test for improper integrals, since

3. is convergent, because is continuous, positive, decreasing and
is convergent by the -test for improper integrals, since

REMARK 1: When we use the Integral Test it is not necessary to start the series or the integral at For instance, in testing the series

REMARK 2: It is not necessary that be always decreasing. It has to be ultimately decreasing,
that is, decreasing for larger than some number

EXAMPLES: Determine whether the following series converge or diverge.

(a)

Solution: The function is continuous, positive and decreasing on therefore
we can apply the Integral Test:

Since this integral diverges, the series also diverges.

(b)

Solution: The function is continuous, positive and decreasing on therefore
we can apply the Integral Test:

Since this integral converges, the series also converges. We also note that, as it was mentioned before,

(c)

Solution: The function is positive and continuous for
However, looking at the graph of this function we conclude that is not decreasing.

At the same time one can show that this function is ultimately decreasing. In fact,

Note that for all sufficiently large which means that and therefore is ultimately decreasing.
So, we can apply the Integral Test:

Since this integral diverges, the series also diverges.

THEOREM ( -Test): The -series is convergent if and divergent if

Proof: We distinguish three cases:

Case I: If then therefore diverges by the Divergence Test.

Case II: If then

therefore diverges by the Divergence Test.

Case III: If then the function is continuous, positive and decreasing on therefore
we can apply the Integral Test by which is convergent if and only if the improper integral
is convergent. But is convergent if and divergent if
by the -test for improper integrals.

REMARK: As before, when we use the -Test it is not necessary to start the series at

EXAMPLES: Determine whether the following series converge or diverge:

(a)

Solution: The series diverges by the -test for series, since

(b)

Solution: The series converges by the -test for series, since

(c)

Solution: The series converges by the -test for series, since

(d)

REMARK: Note that we CAN'T apply the -test directly!

Solution 1: The function is continuous, positive and decreasing on therefore
we can apply the Integral Test:

Since this integral diverges, the series also diverges.

REMARK: Note that we CAN'T apply the -test directly!

Solution 2: We have

Since diverges by the -test with , it follows that
also diverges, since convergence or divergence is unaffected by deleting a finite number of terms.

Comparison Tests

THE COMPARISON TEST: Suppose that and are series with positive terms.

(a) If is convergent and for all then is also convergent

(b) If is divergent and for all then is also divergent

EXAMPLES:
Use the Comparison Test to determine whether the following series
converge or diverge.

(a)

Solution: Since for and
diverges by the -test
with ,
it follows that
also diverges.

(b)

Solution: Since and
diverges by the -test with ,
it follows that
also diverges.

(c)

Solution:
Since

it follows that
also converges.

(d)

Solution:
Since

it follows that
also converges.

(e)

Solution 1:
Since

it follows that
also converges.

Solution 2: Rewrite
as

Since

it follows that and therefore
also converges.

THE LIMIT COMPARISON TEST: Suppose that and are series with positive terms.
If

where is a finite number and then either both series converge or both diverge.

EXAMPLES:
Use the Limit Comparison Test to determine whether the following series
converge or diverge:

(a)

Solution: Put

Then

Since and
converges by the -test with , it follows that
also converges.

(b)

Solution:
Put

Then

Since and
diverges by the -test with , it follows that
also diverges.

(c)

Solution:
Put

Then

Since and
converges by the -test with , it follows that
also converges.

(d)

Solution:
Put

Then

Since and
converges by the -test with , it follows that
also converges.

(e)

Solution: Put

Then

To evaluate we apply L'Hospital's Rule. To this end
we first put Then

We have

Therefore

Since and
diverges by the -test with , it follows that
also diverges.