1. THE INTEGRAL TEST AND ESTIMATES OF SUMS
Example:
Test the series for convergence or divergence.

2. 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.

3. 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.

4. THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM: (Integral Test)
a continuous, positive, decreasing function on [1, inf)
Convergent
Convergent
Divergent
Dinvergent
REMARK:
When we use the Integral Test, it is not necessary to start the series or the integral at n = 1 . For instance, in testing the series

5. THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM: (Integral Test)
a continuous, positive, decreasing function on [1, inf)
Convergent
Convergent
Divergent
Dinvergent
REMARK:
REMARK:
When we use the Integral Test, it is not necessary to start the series or the integral at n = 1 . For instance, in testing the series
Also, it is not necessary that f(x) be always decreasing. What is important is that f(x) be ultimately decreasing, that is, decreasing for larger than some number N.

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. 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?

8. THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM: (Integral Test)
a continuous, positive, decreasing function on [1, inf)
Convergent
Convergent
Divergent
Dinvergent
Example:
P Series:
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. THE INTEGRAL TEST AND ESTIMATES OF SUMS
TERM-102

13. ESTIMATES OF SUMS
This will be joined with alternating test 11.5

15. S = Sn + Rn S ~ Sn ~ THE INTEGRAL TEST AND ESTIMATES OF SUMS
SYLLABUS: Students should know the “Remainder Estimate for the Integral Test”. Example 5a and Example 6 are excluded.
ESTIMATING THE SUM OF A SERIES
S = Sn Rn
S ~ Sn ~
for sufficient large n
Example:
Estimate the sum
How accurate is this estimation?

16. THE INTEGRAL TEST AND ESTIMATES OF SUMS
Bounds for the Remainder in the Integral Test
Convergent by integral test
Error (how good)
good approximation
REMARK:
We can estimate the sum
Example:

17. THE INTEGRAL TEST AND ESTIMATES OF SUMS
REMARK:
We can estimate the sum
ESTIMATING THE SUM OF A SERIES 1 2 3 4 5 10 20 40 50
1000
11000
21000
61000
71000
81000
91000

18. THE INTEGRAL TEST AND ESTIMATES OF SUMS
Example:
Estimate the sum
How accurate is this estimation?

19. 15 22 Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
Facts about: (Harmonic Seris)
1)The harmonic series diverges, but very slowly. 15
the sum of the first million terms is less than 22
the sum of the first billion terms is less than
2) If we delete from the harmonic series all terms
having the digit 9 in the denominator. The resulting
series is convergent.

20. THE INTEGRAL TEST AND ESTIMATES OF SUMS
TERM-112