1. SERIES TESTS Special Series: Question in the exam
Is the series convergent or divergent?
Special Series:
Series Tests
Geometric Series
Harmonic Series
Telescoping Series
Alter Harmonic
p-series
Alternating p-series
Divergence Test
Integral Test
Comparison Test
Limit Compar Test
Ratio Test
Root Test
Alter Series Test

2. ALTERNATING SERIES TEST

3. ALTERNATING SERIES series with positive terms Series Tests
Divergence Test
Integral Test
Comparison Test
Limit Compar Test
Ratio Test
Root Test
Alter Series Test
series with some positive and some negative terms
All
alternating series

4. ALTERNATING SERIES alternating series n-th term of the series
are positive

5. ALTERNATING SERIES alternating series alternating harmonic series
alternating geomtric series
alternating p-series

6. ALTERNATING SERIES convg Example:
THEOREM: (THE ALTERNATING SERIES TEST)
alternating
decreasing
lim = 0
convg
Remark:
The convergence tests that we have looked at so far apply only to series with positive terms. In this section and the next we learn how to deal with series whose terms are not necessarily positive. Of particular importance are alternating series, whose terms alternate in sign.
Example:
Determine whether the series
converges or diverges.

7. ALTERNATING SERIES convg Example:
THEOREM: (THE ALTERNATING SERIES TEST)
alternating
decreasing
lim = 0
convg
Example:
Determine whether the series
converges or diverges.

8. ALTERNATING SERIES convg Example:
THEOREM: (THE ALTERNATING SERIES TEST)
alternating
decreasing
lim = 0
convg
Example:
Determine whether the series
converges or diverges.

9. Absolutely convergent (AC)
Alternating Series, Absolute and Conditional Convergence
the series of absolute values
DEF:
Example:
The series
is called
Absolutely convergent (AC)
Test the series for absolute convergence.
If the series of absolute values is convergent
Also we may say that converges absolutely

10. Absolutely convergent
Alternating Series, Absolute and Conditional Convergence
DEF: IF
Is called
Absolutely convergent
convergent
converges absolutely
Example:
Test the series for absolute convergence.

11. Conditionally convergent (CC) Is called conditionally convergent
Alternating Series, Absolute and Conditional Convergence
DEF:
The series
is called
Conditionally convergent (CC)
If it is convergent but the series of absolute values is divergent
DEF:
Example:
Is called conditionally convergent
Test the series for absolute convergence.
if it is convergent but not absolutely convergent.
REM:
convg
divg

12. Alternating Series, Absolute and Conditional Convergence
Absolutely
convergent
convergent
THM:
convg
convg
THM:
Example:
Determine whether the series
converges or diverges.
The signs change irregularly

13. conditionally convergent Absolutely convergent
Alternating Series, Absolute and Conditional Convergence
conditionally convergent
Absolutely convergent
convergent
divergent

14. SERIES TESTS Special Series: Questions in the exam
Is the series convergent or divergent?
Special Series:
Geometric Series
Harmonic Series
Telescoping Series
Alter Harmonic
p-series
Alternating p-series

15. ALTERNATING SERIES series with positive terms Series Tests
Divergence Test
Integral Test
Comparison Test
Limit Compar Test
Ratio Test
Root Test
Alter Series Test
series with some positive and some negative terms
All
alternating series

16. THE RATIO TEST

17. Ratio Test Example: THE RATIO AND ROOT TESTS be an infinite series
the Ratio Test is inconclusive; that is, no conclusion can be drawn about the convergence or divergence
Example:
Test the series for convergence.

18. THE RATIO AND ROOT TESTS
TERM-132

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

20. THE ROOT TEST

21. Root Test Example: THE RATIO AND ROOT TESTS be an infinite series
Test the series for convergence.

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

23. Sec 11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
TERM-132

24. SERIES TESTS Special Series: Question in the exam
Is the series convergent or divergent?
Special Series:
Series Tests
Geometric Series
Harmonic Series
Telescoping Series
Alter Harmonic
p-series
Alternating p-series
Divergence Test
Integral Test
Comparison Test
Limit Compar Test
Ratio Test
Root Test
Alter Series Test

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

27. THE RATIO AND ROOT TESTS
TERM-101

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