周次: 13 日期: May 20, 2022 节次: 2
In this lecture, the Comparison Test is upgraded to the limit case. In addition, two new tests based on the comparison with respect to the geometric series is given. These are called the Ratio Test and the Root Test, respectively. Note that all the tests are designed for a series with non-negative terms.
§ 11.4 The Limit Comparison Test
Note in this lecture all the series are considered to have all the terms being non-negative.
Recap: Theorem 10. The Comparison Test
-
If there is a convergent series
such thatthen
also converges. -
If there is a divergent series
such thatthen
also diverges.
Ideas behind the Proof.
For case 1, the partial sum
For case 2, the partial sum
Example 1. Applying the Comparison Test
(a)
( Diverges. Compared with Harmonic series
(b)
( Converges. Compared with the Telescoping series
(c)
( Converges. Compared with the geometric series
Remarks
-
The
-series and geometric , telescoping series are frequently been used to be compared with. -
The Comparison Test ONLY WORKS for the series with non-negative terms (or with non-positive terms).
For example, the following series with both positive and negative terms
diverges, although all the terms
. -
That’s been said, if a series has only finite many negative terms, the comparison test is still applicable. For example
The Limit Comparison Test
Based on the experiences built up from the comparison test, we may get the feeling that testing for convergence is largely about recognizing how fast the
Therefore, the concepts of growth rate or order can be applied to the convergence test.
Theorem 11. Limit Comparison Test
Suppose that
-
If
then
and both converge or both diverge. -
If
then
converges. -
If
then
diverges.
Proof.
Example 2. Using the Limit Comparison Test
Which of the following series converges, and which diverge?
b.
c.
Example 3. Does converge?
§11.5 The Ratio and Root Tests
The Ratio Test
For a geometric series
For other series, it appears that we can measure the rate of growth (or decline) by extending the “Ratio Test”.
Theorem 12. The Ratio Test
Let
Then
(a) the series converges if
(b) the series diverges if
(c) the test is inconclusive if
Proof.
Example 1. Applying the Ratio Test
(a)
( Converges )
(b)
( Diverges )
(c)
( Diverges, but inconclusive from the Ratio Test )
The Root Test
The ratio test is convenient for the case when
Guiding Example
Let
This is not a geometric series, and the
As
A test that will answer the question is the Root Test.
Theorem 13. The Root Test
Let
Then
(a) the series converges if
(b) the series diverges if
(c) the test is inconclusive if
Proof.
Example 3. Applying the Root Test
(a)
( Converges )
(b)
( Diverges )
(c)
( Converges )
Guiding Example (Revisited)
Let
( Converges )