9.4 Comparison of series

1. 9.4 Comparison of series Greg Kelly, Hanford High School, Richland, Washington

2. “Does this series converge, and if so, for what values of x does it converge?” Convergence The series that are of the most interest to us are those that converge. Today we will consider the question:

3. Direct Comparison Test This series converges. For non-negative series: So this series must also converge. If every term of a series is less than the corresponding term of a convergent series, then both series converge. So this series must also diverge. If every term of a series is greater than the corresponding term of a divergent series, then both series diverge. This series diverges.

4. Determine if the series converges or diverges

5. Determine if the series converges or diverges This series very much resembles a special series: which is a Geometric series that converges Every term of our original series is positive and smaller (has a larger denominator) than the geometric series. This series must converge due to the direct comparison test.

6. Does this series converge or diverge?

7. Does this series converge or diverge? This series resembles the divergent series: unfortunately, we can not use the direct Comparison test with this series because each term of this series is smaller than the one that we are looking at

8. Does this series converge or diverge? We will compare to Which is also a divergent P series Comparing terms we find that Therefore this series diverges

9. An Interesting Fact We will prove this fact later in the chapter. However, we will need this fact today in order to doing comparing.

10. is the Taylor series for , which converges. The original series converges. Ex. 3: Prove that converges for all real x. There are no negative terms: larger denominator The direct comparison test only works when the terms are non-negative.

11. If converges, then we say converges absolutely. If converges, then converges. Absolute Convergence The term “converges absolutely” means that the series formed by taking the absolute value of each term converges. Sometimes in the English language we use the word “absolutely” to mean “really” or “actually”. This is not the case here! If the series formed by taking the absolute value of each term converges, then the original series must also converge. “If a series converges absolutely, then it converges.”

12. Since , converges to converges by the direct comparison test. Since converges absolutely, it converges. Ex. 4: We test for absolute convergence:

13. Homework p.630 1-14 all • Two mathematicians are studying a convergent series. The first one says: "Do you realize that the series converges even when all the terms are made positive?" The second one asks: "Are you sure?" "Absolutely!"

14. 1 There is a positive number R such that the series diverges for but converges for . The series converges for every x. ( ) 2 3 The series converges for at and diverges everywhere else. ( ) There are three possibilities for power series convergence. The series converges over some finite interval: (the interval of convergence). The series may or may not converge at the endpoints of the interval. (As in the previous example.) The number R is the radius of convergence.