1. IF Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS DEF:
REMARK: IF
Notice that if a series with positive terms, then absolute convergence is the same as convergence.
Is called
Absolutely
convergent
convergent
Example:
Example:
Test the series for absolute convergence.
Test the series for absolute convergence.

2. Is called conditionally convergent
Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
Example:
DEF:
Test the series for absolute convergence. IF
Is called
Absolutely
convergent
convergent
Example:
DEF:
Test the series for absolute convergence.
Is called conditionally convergent
if it is convergent but not absolutely convergent.

3. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
Absolutely
convergent
convergent
THM:
Example:
Determine whether the series
converges or diverges.

4. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
Absolutely
convergent
convergent
THM:

5. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
Series Tests
Test for Divergence
Integral Test
Comparison Test
Limit Comparison Test
Alternating Test
Ratio Test
Let

6. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
Ratio Test
Let
Example:
Example:
Test the series for absolute convergence.
Test the series for absolute convergence.

7. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
Ratio Test
Let
REMARK:
Case L = 1 means that the test gives no information.

8. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
TERM-101

9. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
Series Tests
Example:
Test for Divergence
Integral Test
Comparison Test
Limit Comparison Test
Alternating Test
Ratio Test
Test the series for convergence.
Root Test
Let

10. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
TERM-082

11. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
REARRANGEMENTS
Divergent
If we rearrange the order of the terms in a finite sum, then of course the value of the sum remains unchanged.
But this is not always the case for an infinite series.
By a rearrangement of an infinite series we mean a series obtained by simply changing the order of the terms.

12. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
REARRANGEMENTS
Divergent
convergent
See page 719

13. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
REARRANGEMENTS
REMARK:
Absolutely
convergent
any rearrangement has the same sum s
with sum s
Riemann proved that
Conditionally
convergent
there is a rearrangement that has a sum equal to r.
r is any real number

14. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
TERM-091

15. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
TERM-082

16. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
TERM-082

17. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
TERM-091