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 either both converge or both diverge. Ex. 1 Applying the Integral Test Does converge?

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 This is probably the most famous divergent series in mathematics. series.

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? 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?

4 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

5 Ex. 4 Using the Limit Comparison Test Determine whether each series converge or diverge. a) b) c) d)

6 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)

7 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.

8 Absolute and Conditional Convergence The harmonic series is convergent, but not absolutely convergent, therefore we say that the harmonic series is We can rearrange the terms of a finite sum without affecting the sum. We can also do this for a finite number of terms of an infinite series, without affecting the sum. We can also do this for an infinite number of terms of an infinite series, leaving the sum unaltered, ONLY if the series converges absolutely. 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 and Rearrangements of conditionally convergent series Alternatively, if Ʃa n converges conditionally, then the terms can be rearranged to form a divergent series. The terms can also be rearranged to form a series that converges to any preassigned sum.

9 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.

10 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.

11 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) b) c)