Stewart Calculus Section 11.4

1. 11.4 The Comparison Tests
The convergence of some series can be
determined easily by the integral test, for
example:
=
Some series look similar, but the integral test
cannot be easily applied, for example:
=
+

2. The Comparison Test:
∞ ∞
Suppose =
and = are series with
positive terms,
∞
If = is convergent and ≤
∞
then = is also convergent.
∞
If = is divergent and ≥
∞
then = is also divergent.
If = > is finite
then they either both converge or both diverge.

3. Common series used for comparison:
p-series: converges for >
=
diverges for
=
geometric series: converges for | | <
=
diverges for | |
=

4. Ex: Test the series for convergence.
=
+

5. Ex: Test the series for convergence.
=
+
<
+
is convergent.
=
So is convergent.
=
+

6. Ex: Test the series for convergence.
=

7. Ex: Test the series for convergence.
=
>
is divergent.
=
So is divergent.
=

8. Ex: Test the series for convergence.
=

9. Ex: Test the series for convergence.
=
=
is convergent.
=
So is convergent.
=

10. +
Ex: Test the series for convergence.
=
+

11. +
Ex: Test the series for convergence.
=
+
+
=
+
is divergent (p-test).
=
+
So is divergent.
=
+

12. +
Ex: Test the series for convergence.
=
+

13. +
Ex: Test the series for convergence.
=
+
+
=
+
is convergent (geometric).
=
+
So is convergent.
=
+