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Консультация № 182630
26.03.2011, 20:04
72.45 руб.
27.03.2011, 00:50
0
1
1
Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Найдите область сходимости функционального ряда:
а) (n^3+1)/(3^n*(x-2)^n) В числителе - эн в кубе плюс один, в знаменателе - 3 в степени эн, умноженное на (х-2) в степени эн. Впереди дроби - сигма, эн пробегает значения от 1 до бесконечности)
б) n!/x^n (эн факториал, делённый на икс в степени эн. Впереди дроби - сигма, эн пробегает значения от 1 (под сигмой) до бесконечности (над сигмой))
Найдите область сходимости функционального ряда:
а) (n^3+1)/(3^n*(x-2)^n) В числителе - эн в кубе плюс один, в знаменателе - 3 в степени эн, умноженное на (х-2) в степени эн. Впереди дроби - сигма, эн пробегает значения от 1 до бесконечности)
б) n!/x^n (эн факториал, делённый на икс в степени эн. Впереди дроби - сигма, эн пробегает значения от 1 (под сигмой) до бесконечности (над сигмой))
Обсуждение
27.03.2011, 06:58
общий
это ответ
Здравствуйте, Aleksandrkib!
а) Воспользуемся признаком Даламбера:
Отсюда |x-2|>1/3, то есть ряд сходится при x<5/3 и при x>7/3. Исследуем сходимость ряда на границе. При x=7/3 имеем ряд
который, очевидно, расходится. При x=5/3 имеем знакочередующийся ряд
который также расходится. Следовательно, область сходимости исходного ряда - (-[$8734$],5/3)[$8746$](7/3,[$8734$]).
б) Для этого ряда
то есть не выполняется необходимое условие сходимости. Следовательно, ряд расходится при всех конечных x.
а) Воспользуемся признаком Даламбера:
Отсюда |x-2|>1/3, то есть ряд сходится при x<5/3 и при x>7/3. Исследуем сходимость ряда на границе. При x=7/3 имеем ряд
который, очевидно, расходится. При x=5/3 имеем знакочередующийся ряд
который также расходится. Следовательно, область сходимости исходного ряда - (-[$8734$],5/3)[$8746$](7/3,[$8734$]).
б) Для этого ряда
то есть не выполняется необходимое условие сходимости. Следовательно, ряд расходится при всех конечных x.
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