tailieunhanh - Handbook of mathematics for engineers and scienteists part 54

Handbook of mathematics for engineers and scienteists part 54. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | . Numerical Series and Infinite Products 339 If D 1 then the series 22 an is convergent. If D 1 then the series is divergent. For n 1 D 1 the D Alembert criterion cannot be used for deciding whether the series is convergent or divergent. Example 1. Let us examine convergence of the series nk xn with x 0 using the D Alembert criterion. n 1 Taking an nk xn we get k an 1 t d 1 --- 11 I x x as n . Therefore D x. It follows that the series is convergent for x 1 and divergent for x 1. 4. Cauchy s criterion. Suppose that there exists the limit finite or infinite lim K. n x For K 1 the series 22 an is convergent for K 1 the series is divergent. For K 1 n 1 the Cauchy criterion cannot be used to establish convergence of a series. Remark. The Cauchy criterion is stronger than the D Alembert criterion but the latter is simpler than the former. 5. Gauss criterion. Suppose that the ratio of two consecutive terms of a series can be represented in the form an A o f as n to. an 1 n n tt The series 22 an is convergent if A 1 or A 1 1. The series is divergent if A 1 or n 1 A 1 1. 6. Maclaurin-Cauchy integral criterion. Let f x be a nonnegative nonincreasing continuous function on the interval 1 x to. Let f 1 a1 f 2 a2 . f n an . Then the series an is convergent if and only if the improper integral i f x dx is n 1 1 convergent. Example 2. The harmonic series V 1 x- is divergent since the integral dx is n 1 n 2 3 J1 x divergent. In a similar way one finds that the series a is convergent for a 1 and divergent for a 1. n 1 . Other criteria of convergence divergence of series with positive terms. 1. Raabe criterion. Suppose that there exists the limit finite or infinite u an 1A -n lim n----------1 R. n - an 1 J Then for R 1 the series 22 an is convergent and for R 1 it is divergent. For R 1 the Raabe criterion is inapplicable. 340 Series 2. Bertrand criterion. Suppose that there exists the limit finite or infinite lim In n n œ B. tt Then for B 1 the series 52 an is convergent .