9 5 Testing Convergence at Endpoints

Transcription

1 9 5 Testing Convergence at Endpoints In this section we will investigate convergence using more tests. We will also looks at three specific types of series. The Integral Test Let {a n } be a sequence of positive terms. Suppose that a n = f(n) where f is a continuous, positive decreasing function of x x N (N is a positive integer). Then the series and the integral diverge. either both converge or both Ex. 1 Applying the Integral Test Does converge? Feb 16 8:39 PM 1

2 Harmonic and p series The integral test can be used to determine convergence for any series of the form p a real constant. These series are known as the Explore 1) Use the integral test to prove that converges if p>1. 2) Use the integral test to prove that diverges if p<1. 3) Use the integral test to prove that diverges if p=1. The p series with p=1 is the series. This is probably the most famous divergent series in mathematics. Feb 16 8:42 PM 2

3 Ex.2 Find the first 5 partial sums of How many terms do you think it would take for the sum form a value bigger than 2.676? Feb 16 8:47 PM 3

4 Ex. 3 The Slow Divergence of the Harmonic Series Approximately how many terms of the harmonic series are required to form a partial sum larger than 30? Feb 16 8:47 PM 4

5 LIMIT COMPARISON TEST Suppose that a n >0 and b n >0 n>n (N is a positive integer) 1) If, 0< c <, then Ʃa n and Ʃb n 2) If and Ʃb n converges, then Ʃa n 3) If, and Ʃb n diverges, then Ʃa n Feb 16 8:49 PM 5

6 Ex. 4 Using the Limit Comparison Test Determine whether each series converge or diverge. a) b) c) d) Feb 16 8:55 PM 6

7 Alternating Series A series in which the terms are alternately positive and negative is known as an e.g. Alternating Series Test aka Leibniz s Theorem The series converges if all three of the following conditions are satisfied: 1) 2) 3) Feb 16 8:57 PM 7

8 The Alternating Series Estimation Theorem If the alternating series satisfies the conditions of Leibniz s theorem the truncation error for the nth partial sum is less than u n+1 and has the same sign as the first unused term. Ex. 5 The Alternating Harmonic Series Prove that the alternating harmonic series is convergent, but not absolutely convergent. Find a bound for the truncation error after 99 terms. Feb 16 9:01 PM 8

9 Absolute and Conditional Convergence The alternating harmonic series is convergent, but not absolutely convergent, therefore we say that the alternating harmonic series is Rearrangements of absolutely convergent series If Ʃa n converges absolutely, and if b 1, b 2, b 3,...,b n,... is any rearrangement of the sequence {a n }, then {b n } converges absolutely. Rearrangements of conditionally convergent series We can rearrange a conditionally convergent series and make it look convergent or divergent. Feb 16 9:09 PM 9

10 Ex. 6 Rearranging the Alternating Harmonic Series Show how to arrange the terms of to form (i) a divergent series (ii) a series that converges to e. Feb 16 9:15 PM 10

11 Intervals of Convergence How do we test a power series for convergence? Here are three steps to help. 1) Use the Ratio test to find the values of x for which the series converges absolutely. Ordinarily, this is an open interval a R< x <a+r. In some instances, the series converges for all values of x. In rare cases, the series converges only at x=a. 2) If the interval of absolute convergence is finite, test for convergence or divergence at each endpoint. The ratio test fails at these points. Use a comparison test, the integral test or the alternating series test. 3) If the interval of absolute convergence is a R< x <a+r, conclude that the series diverges (it does not converge conditionally) for x a >R, because for those values of x the nth term does not approach zero. Feb 16 9:17 PM 11

12 Ex. 7 Finding intervals of convergence Write out the first 4 terms of the following series. For what values of x do the following series converge? a) c) b) d) Feb 16 9:21 PM 12