Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 55', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 346 Series Remark. The Kummer transformation is used for accelerating convergence of series since the generic term a 1 - -k an of the transformed series tends to zero faster that the generic term an of the original series lim aif an 0. The auxiliary sequence 6 is chosen in such a way that n the sum is known beforehand. 2 . Abel transformation x x 2 anbn Ypn - fln 1 Bra n Bn bk Here it is assumed that the sequence Bn is bounded and lim an 0 for example these . . -c. r . . . . . n X conditions are satisfied if the sequence an has finitely many nonzero terms . Remark. The Abel transformation is used for accelerating convergence of series whose convergence is slow if lim an 1 an 1 and bn 0 1 . n œ . Infinite Products . Convergent and divergent infinite products. An infinite product is an expression of the form U1U2U3 H an an 0 where an are real in the general case complex numbers. The expression n Pn aa an ak k 1 is called a finite product. One says that an infinite product is convergent to p if there exists a finite nonzero limit of the partial products n lim n ak p. n A A k 1 If there is no such limit or p to or p 0 one says that the infinite product is divergent. In the last case one says that the infinite product diverges to zero p 0 . tt The simplest examples of infinite products ff an n 1 for an 1 the infinite product is convergent pn 1 for an -1 n the infinite product is divergent Pn has no limit for an n the infinite product is divergent Pn œ for an 1 n the infinite product is divergent to zero Pn 0. . Numerical Series and Infinite Products 347 . Infinite products with positive factors. Taking logarithm of relation one can reduce the problem of convergence of infinite products with positive factors an to the problem of convergence of partial sums of numerical series n Sn ln Pn ln ak. Therefore in order to examine convergence of infinite products one can use the convergence criteria for .
