MATH 155 SUPPLEMENTAL NOTES 22

COMPARISION TEST FOR SERIES OF NONNEGATIVE TERMS

So far in these supplemental notes we have determined that the following series converge:

 

Geometric series with | r | < 1

Telescoping series

p-series when p > 1

and that the following series diverge:

 

Geometric series with | r | ³ 1

p-series when p £ 1.

We have also learned how to use the integral test to determine the convergence or divergence of a series. Now we are going to explore two more methods for determining the convergence or divergence of a series. The two new methods are the Direct Comparison Test and the Limit Comparison Test.

THE DIRECT COMPARISON TEST

DIRECT COMPARISON TEST FOR SERIES OF NONNEGATIVE TERMS

Let S a n be a series with no negative terms.

a.

S a n converges if there is a convergent series S c n with a n £ c n for all n > N, for some integer N.

b.

S a n diverges if there is a divergent series of nonnegative terms S d n with a n > d n for all n > N, for some integer N.

EXAMPLE 1:

Does the following series converge or diverge?

SOLUTION:

EXAMPLE 2:

Does the following series converge or diverge?

SOLUTION:

EXAMPLE 3:

Does the following series converge or diverge?

SOLUTION:

 

EXAMPLE 4:

Does the following series converge or diverge?

 SOLUTION:

 

The main drawback with this test is coming up with a function that satisfies the conditions of this test. Sometimes it would be nice to use the end behavior model for the comparison test, but the inequality might not be satisfied. There has to be an easier way, and it is the limit comparison test.

THE LIMIT COMPARISON TEST

THE LIMIT COMPARISON TEST

Suppose that a n > 0 and b n > 0 for all n ³ N (N an integer).

EXAMPLE 5:

Does the following series converge or diverge?

SOLUTION:

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:

Because ln n grows more slowly than any n c, (ln n) 3 will grow more slowly than any n 3c. Letting c = 1 will give us the following conclusion.

(Another reason why I bounded the nth term of this series this way, is the fact that I would like to have an n left in the denominator of the expression a n / b n.)

EXAMPLE 9:

Does the following series converge or diverge?

SOLUTION:

I have worked several examples using both tests discussed in this set of supplemental notes. Some of these series required some creative thinking to come up with a series to compare them with. That is one of the main drawbacks of any comparison test. Work through these examples. Try to understand the reasons why I approached the problem the way that I did. If you have any questions or problems with any of these examples, please feel free to contact me.

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