1. Chapter 9.4
COMPARISONS OF SERIES

2. After you finish your HOMEWORK you will be able to…
Us e the Direct Comparison Test to determine whether a series converges or diverges
Use the Limit Comparison Test to determine whether a series converges or diverges.

3. THEOREM 9.12 DIRECT COMPARISON TEST
Condition: Let
1. If converges, then converges.
If diverges, then diverges.

4. USING THE DIRECT COMPARISON TEST
Determine the convergence or divergence of
Hmm…what series looks similar to this one?

5. You got it! It is similar to the geometric series
Now we need to check if
For n = 1, we get ,
n = 2 we get Can we use the test?

6. Yes!
Conclusion:
Since is a geometric series with
which fulfills the condition ,
it converges (Thm. 9.6), so it follows
that also converges by the direct
comparison test (Thm. 9.12).

7. THEOREM 9.13 LIMIT COMPARISON TEST
Condition: Let
Where is finite and positive. Then the two
series and either both converge or
both diverge.