Convergence of Series

1. Photo by Vickie Kelly, 2008 Greg Kelly, Hanford High School, Richland, Washington Convergence of Series Abraham Lincoln’s Home Springfield, Illinois

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. nth term test for divergence diverges if fails to exist or is not zero. If then grows without bound. If then As , eventually is larger than , therefore the numerator grows faster than the denominator. The series diverges. (except when x=0) The first requirement of convergence is that the terms must approach zero. Ex. 2: Note that this can prove that a series diverges, but can not prove that a series converges.

4. 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 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.

5. 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.

6. 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.

7. 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.”

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

9. where r = common ratio between terms When , the series converges. Ratio Technique We have learned that the partial sum of a geometric series is given by:

10. For , if then: if the series converges. if the series diverges. if the series may or may not converge. Geometric series have a constant ratio between terms. Other series have ratios that are not constant. If the absolute value of the limit of the ratio between consecutive terms is less than one, then the series will converge.

11. Ex: If we replace x with x-1, we get: If the limit of the ratio between consecutive terms is less than one, then the series will converge.

12. The interval of convergence is (0,2). The radius of convergence is 1. If the limit of the ratio between consecutive terms is less than one, then the series will converge.

13. Ex:

14. The interval of convergence is (2,8). The radius of convergence is . Ex:

15. Ex:

16. Radius of convergence = 0. for all . At , the series is , which converges to zero. Ex: Note: If R is infinite, then the series converges for all values of x.

17. Telescoping Series converges to Another series for which it is easy to find the sum is the telescoping series. Using partial fractions: Ex. 6: p