Math 409-502Harold P. [email protected] of convergence tests• If an6→ 0, then∑nandiverges.• Comparison tests for positive series: if an≤ bnfor all large n, or alternatively if an/bnhasa finite limit, then convergence of∑nbnimplies convergence of∑nan.• Absolute convergence implies convergence: if∑n|an| converges, then so does∑nan.• Ratio and root tests: if either limn→ ∞|an|1/nor limn→ ∞|an+1/an| exists and is strictly less than 1,then∑nanconverges.• Special tests for decreasing positive terms an:(i) if f (x) ↓ 0 as x → ∞, thenR∞f (x) dx and∑∞f (n) have the same conver-gence/divergence behavior (integral test); (ii) if an↓ 0, then∑n(−1)nanconverges (al-ternating series test).Math 409-502 October 6, 2004 — slide #2Cauchy’s condensation testAnother special test for decreasing termsSuppose 0 < an+1≤ anfor all (large) n. Then the two series∑nanand∑n2na2neither bothconverge or both diverge.Example:∑n1n ln(n)Since1n ln(n)is a decreasing function of n, the test applies and says that the conver-gence/divergence behavior is the same as for the series∑n2n12nln(2n). That simplifies to∑n1n ln(2), which is a multiple of the divergent harmonic series. Therefore the original series∑n1n ln(n)diverges too.Math 409-502 October 6, 2004 — slide #3Proof of the condensation test (sketch)a8+ a9+ ···+ a15≤ 8a8≤ 2(a4+ a5+ a6+ a7)a16+ a17+ ···+ a31≤ 16a16≤ 2(a8+ a9+ ···+ a15)...Adding such inequalities shows that partial sums of∑n2na2nare bounded below by partialsums of∑nanand are bounded above by twice the partial sums of∑nan. Therefore the twoseries have the same convergence/divergence behavior.Math 409-502 October 6, 2004 — slide #4Power seriesExample:∞∑n=1xn2n√nFor which values of x does that series converge?[This is Exercise 8.1/1a on page 123.]Solution:By the root test, the series converges (absolutely) when1 > limn→ ∞¯¯¯¯xn2n√n¯¯¯¯1/n= limn→ ∞|x|2√n1/n=|x|2,that is, when |x| < 2.The series diverges when |x| > 2 by the (proof of the) root test.A different test is needed to see what happens when x = ±2.Math 409-502 October 6, 2004 — slide #5Homework• Read section 8.1, pages 114–117.• Do Exercise 7.6/1a,c on page 111.• Do Exercise 8.1/1g on page 123.Math 409-502 October 6, 2004 — slide