Lecture 24Section 11.4 Absolute and Conditional

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1 Lecture 4Section.4 Absolute and Conditional Convergence; Alternating Series Jiwen He Convergence Tests Basic Series that Converge or Diverge Basic Series that Converge Geometric series: p-series: x, if x < p, if p > Basic Series that Diverge Any series a for which lim a 0 p-series: p, if p Convergence Tests () Basic Test for Convergence Keep in Mind that, if a 0, then the series a diverges; therefore there is no reason to apply any special convergence test. Examples. x with x (e.g, ( ) ) diverge since x 0. [ex] diverges since [ex] ( ) diverges since a = ( ) e 0. Convergence Tests () Comparison Tests Rational terms are most easily handled by basic comparison or limit comparison with p-series / p Basic Comparison Test

2 3 + converges by comparison with converges by comparison with 3 converges by comparison with diverges by comparison with 3( + ) comparison with + 6 diverges by ln( + 6) Limit Comparison Test 3 converges by comparison with diverges + by comparison with converges by comparison with 5 Convergence Tests (3) Root Test and Ratio Test The root test is used only if powers are involved. Root Test converges: (a ) / = [ /] (ln ) converges: (a ) / = ln 0 ( ) ( ) converges: (a ) / = + ( ) e Convergence Tests (4) Root Test and Ratio Test The ratio test is effective with factorials and with combinations of powers and factorials. Ratio Comparison Test converges: a + a = (+) 0 converges: a + 3 converges: a + a + 0 a = 0 + 0! converges: a + a = + 0 = (/3) 3 (/3) 3 Absolute Convergence. Absolute Convergence Absolute Convergence! diverges: a + a = ( ) + e converges:! a + a =

3 Absolute Convergence A series a is said to converge absolutely if a converges. if a converges, then a converges. i.e., absolutely convergent series are convergent. Alternating p-series with p > ( ) p, p >, converge absolutely because p converges. ( ) + = = + 3 converge absolutely. 4 Geometric Series with < x < ( ) j() x, < x <, converge absolutely because x converges converge absolutely. 6 Conditional Convergence Conditional Convergence A series a is said to converge conditionally if a converges while a diverges. Alternating p-series with 0 < p ( ) p, 0 < p, converge conditionally because p diverges. ( ) + = + 3 converge conditionally. 4 = 3 Alternating Series Alternating Series Alternating Series Let {a } be a sequence of positive numbers. ( ) a = a 0 a + a a 3 + a 4 is called an alternating series. Alternating Series Test Let {a } be a decreasing sequence of positive numbers. If a 0, then ( ) a converges. Alternating p-series with p > 0 ( ) p, p > 0, converge since f(x) = x p is decreasing, i.e., f (x) = p x p+ > 0 for x > 0, and lim f(x) = 0. ( ) + = x = converge conditionally. 3

4 Examples ( ), converge since f(x) = + x + is decreasing, i.e., f (x) = (x + ) > 0 for x > 0, and lim f(x) = 0. x ( ) x, converge since f(x) = + 0 x + 0 is decreasing, i.e., f (x) = x 0 (x + 0) > 0, for x > 0, and lim f(x) = 0. x An Estimate for Alternating Series An Estimate for Alternating Series Let {a } be a decreasing sequence of positive numbers that tends to 0 and let L = ( ) a. Then the sum L lies between consecutive partial sums s n, =0 s n+, s n < L < s n+, if n is odd; s n+ < L < s n, if n is even. and thus s n approximates L to within a n+ Example Find s n to approximate Set ( ) + = = =0 L s n < a n+. ( ) + = + 3 within 0. = ( ) +. For L s n < 0, we want a n+ = (n + ) + < 0 n + > 0 n > 98. Then n = 99 and the 99th partial sum s 00 is s 99 = From the estimate we conclude that s < L s 99 < a 00 = ( ) + = ln < s 00 = 4

5 Example Find s n to approximate =0 ( ) + ( + )! = 3! + 5! within 0. For L s n < 0, we want a n+ = ((n + ) + )! < 0 n. Then n = and the nd partial sum s is From the estimate we conclude that s < s = 3! L s < a = 5! =0 4 Rearrangements ( ) + ( + )! = sin < s Why Absolute Convergence Matters: Rearrangements () Rearrangement of Absolute Convergence Series ( ) = = 3 absolutely =0 Rearrangement ? = = 3 Theorem. All rearrangements of an absolutely convergent series converge absolutely to the same sum. Why Absolute Convergence Matters: Rearrangements () Rearrangement of Conditional Convergence Series ( ) + = = ln conditionally 6 = Rearrangement ? = ln Multiply the original series by ( ) + = = ln = Adding the two series, we get the rearrangement ( ) + + ( ) + = = 3 ln = Remar = 5

6 A series that is only conditionally convergent can be rearranged to converge to any number we please. It can also be arranged to diverge to + or, or even to oscillate between any two bounds we choose. Outline Contents Convergence Tests Absolute Convergence. Absolute Convergence Alternating Series 3 4 Rearrangements 5 6