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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 5

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 2 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | We bound the tail of the series of an . E a E a tt s Ỉrn rN 1 r v o an is absolutely convergent. Example 12.1.8 Consider the series E nabn n 0 where a and b are real constants. We use the root test to check for absolute convergence. lim nabn 1 n 1 n -tt m lim na n 1 n b exp lim lllin 1 n n ỉ b e0 1 b 1 Thus we see that the series converges absolutely for b 1. Note that the value of a does not affect the absolute convergence. Example 12.1.9 Consider the absolutely convergent series E zỊ2 n2 534 We aply the root test. lim an 1 n lim n -tt n -tt 1 1 n n2 lim n -2 n n -tt lim e-2 ln n n e0 1 It fails to predict the convergence of the series. Raabe s Test Result 12.1.5 The series 2 an converges absolutely if lim n 1 - a 1. n - y an If the limit is less than unity then the series diverges or converges conditionally. If the limit is unity the test fails. .