When given a series S a _n, there are two questions you should ask yourself. The first is does this series converge or diverge? The second is if it converges, what is its sum? Let us deal with the first question first.

all terms are greater than or equal to zero is called a series of nonnegative terms. We could look at the sequence of partial sums and determine if they form a non-decreasing sequence. Then we could use the non-decreasing sequence theorem to determine the convergence of this sequence. This theorem is talked about in supplemental notes 18. But as we discussed before, we will have to look at a lot of terms of this sequence to determine that it remains non-decreasing and bounded from above. So there has to be an easier way.

THE INTEGRAL TEST

Let {a _n} be a sequence of positive terms. Suppose that a _n = f (n), where f is a continuous, positive, decreasing function of x for x ³ N (N a positive integer).

I will do three examples illustrating this method, and then I will make a very important point.

EXAMPLE 1:

Does the following series converge or diverge?

SOLUTION:

The improper integral diverges, therefore by the integral test the series diverges.

This series has a name, it is called the harmonic series and it will always diverge.

EXAMPLE 2:

Does the following series converge or diverge?

SOLUTION:

The improper integral converges, therefore by the integral test the series converges.

EXAMPLE 3:

Does the following series converge or diverge?

SOLUTION:

The improper integral diverges, therefore by the integral test the series diverges.

The last three examples featured series that belong to a group of series called p-series, and here is a basic fact about all p-series.

FACT:

Now, if you see any series of this form, look at the exponent and determine what case the p-series falls into. That will determine its convergence or divergence.

EXAMPLE 4:

Does the following series converge or diverge?

SOLUTION:

The improper integral converges, therefore by the integral test the series converges.

EXAMPLE 5:

Does the following series converge or diverge?

SOLUTION:

The improper integral diverges, therefore by the integral test the series diverges.

EXAMPLE 6:

Does the following series converge or diverge?

SOLUTION:

EXAMPLE 7:

Does the following series converge or diverge?

SOLUTION:

EXAMPLE 8:

Does the following series converge or diverge?

SOLUTION:

The improper integral converges, therefore by the integral test the series converges.