1. Tests for Convergence, Pt. 1
The comparison test,
the limit comparison test,
and the integral test.

2. Comparing series. . .
Consider two series ,
with for all k.
In this presentation, we will base all our series at 1, but similar results apply if they start at 0 or elsewhere.

3. Comparing series. . . Note that: What does this tell us?
Consider two series ,
with for all k.
How are these related in terms of convergence or divergence?
Note that:
What does this tell us?

4. Comparing series. . . Note that:
Consider two series ,
with for all k.
Note that:
Where does the fact that the terms are non-negative come in?
What does this tell us?

5. Series with positive terms. . .
Since for all positive integers k,
So the sequence of partial sums is . . .
Non-decreasing Bounded above Geometric

6. Suppose that the series converges
So the sequence of partial sums is . . .
Non-decreasing Bounded above Geometric

7. The two “ingredients” together. . .
Partial sums
are
non-decreasing
Terms of a series
are non-negative
Partial sums of
are
bounded above.
and
converges

8. A variant of a familiar theorem
Suppose that the sequence is non-decreasing and bounded above by a number A. That is, . . .
Theorem 3 on page 553 of OZ
Then the series converges to some value that
is smaller than or equal to A.

9. This gives us. . . The Comparison Test:
Suppose we have two series , with
for all positive integers k.
If converges, so does and
If diverges, so does

10. This test is not in the book!
A related test. . .
This test is not in the book!
There is a test that is closely related to the comparison test, but is generally easier to apply. . . It is called the
Limit Comparison Test

11. (One case of…) The Limit Comparison Test
Limit Comparison Test: Consider two series
with , each with positive terms.
If , then
are either both convergent or both divergent.
Why does this work?

12. (Hand waving) Answer: Because if
Then for “large” n, ak  t bk.This means that “in the long run” the partial sums
behave similarly in terms of convergence or divergence.

13. The Integral Test
y = a(x)
Suppose that we have a sequence {ak} and we associate it with a continuous function
y = a(x), as we did a few days ago. . .
Look at the graph. . .
What do you see?

14. The Integral Test So converges diverges If the integral
y = a(x)
converges
diverges
If the integral
so does the series.

15. The Integral Test Now look at this graph. . . What do you see?
y = a(x)
Now look at this graph. . .
What do you see?

16. The Integral Test So converges diverges If the integral
y = a(x)
Why 2?
converges
diverges
If the integral
so does the series.

17. The Integral Test The Integral Test:
Suppose for all x  1, the function a(x) is continuous, positive, and decreasing. Consider the series
and the integral
If the integral converges, then so does the series.
If the integral diverges, then so does the series.