1. 3 TESTS Sec 11.3: THE INTEGRAL TEST Sec 11.4: THE COMPARISON TESTS
Sec 11.4: THE LIMIT COMPARISON TESTS

2. Sec 11.3:
THE INTEGRAL
TEST

3. Sec 11.3: THE INTEGRAL TEST Example:
Test the series for convergence or divergence.

4. THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM: (Integral Test)
Remark:
a continuous, positive, decreasing function on [1, inf)
Convergent
Convergent
sequence of positive terms.
Divergent
Dinvergent
Example:
Solution:
Test the series for convergence or divergence.
Since this improper integral is convergent, the series is also convergent by the integral test.

5. THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM: (Integral Test)
a continuous, positive, decreasing function on [1, inf)
Convergent
Convergent
Divergent
Dinvergent
Example:
Solution:
Test the series for convergence or divergence.
Since this improper integral is divergent, the series is also divergent by the integral test.

6. THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM: (Integral Test)
a continuous, positive, decreasing function on [1, inf)
Convergent
Convergent
Divergent
Dinvergent
Special Series:
Example:
Geometric Series
Harmonic Series
Telescoping Series
p-series
Alternatingp-series
Harmonic Series
is the series convergent?

7. THE INTEGRAL TEST AND ESTIMATES OF SUMS
TERM-102

8. Memorize: THE INTEGRAL TEST AND ESTIMATES OF SUMS Example:
Special Series:
Geometric Series
Harmonic Series
Telescoping Series
p-series
Alternatingp-series
Example:
For what values of p is the series convergent?

9. THE INTEGRAL TEST AND ESTIMATES OF SUMS
Example:
P Series:
For what values of p is the series convergent?
Example:
Example:
Test the series for convergence or divergence.
Test the series for convergence or divergence.

10. THE INTEGRAL TEST AND ESTIMATES OF SUMS
FINAL-081

11. THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM: (Integral Test)
a continuous, positive, decreasing function on [1, inf)
Convergent
Convergent
Divergent
Dinvergent
REMARK:
REMARK:
We should not infer from the Integral Test that the sum of the series is equal to
the value of the integral. In fact,
Integral Test just test if convergent or divergent. But if it is convergent what is the sum??

12. Sec 11.4:
THE COMPARISON TESTS

13. Sec 11.4: THE COMPARISON TESTS
In the comparison tests the idea is to compare a given series with a series that is known to be convergent or divergent.
THEOREM: (THE COMPARISON TEST)
convg
Known Series
Example:
Determine whether the series
converges or diverges.
geometric
Solution:
P-series

14. Sec 11.4: THE COMPARISON TESTS
In the comparison tests the idea is to compare a given series with a series that is known to be convergent or divergent.
THEOREM: (THE COMPARISON TEST)
convg
Known Series
Example:
Determine whether the series
converges or diverges.
geometric
Solution:
P-series

15. Sec 11.4: THE COMPARISON TESTS
In the comparison tests the idea is to compare a given series with a series that is known to be convergent or divergent.
THEOREM: (THE COMPARISON TEST)
convg
THEOREM: (THE COMPARISON TEST)
divg

16. Sec 11.4: THE COMPARISON TESTS
In the comparison tests the idea is to compare a given series with a series that is known to be convergent or divergent.
THEOREM: (THE COMPARISON TEST)
divg
Example:
Determine whether the series
converges or diverges.
Solution:

17. THE LIMIT COMPARISON TESTS
Sec 11.4:
THE LIMIT COMPARISON TESTS

18. Sec 11.4: THE COMPARISON TESTS
THEOREM: (THE LIMIT COMPARISON TEST)
either both series converge or both diverge.
With positive terms
Example:
Example:
Solution:
Solution:
Determine whether the series converges or diverges.
Determine whether the series converges or diverges.
REMARK:
Notice that in testing many series we find a suitable comparison series by keeping only the highest powers in the numerator and denominator.