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Консультация № 194535
28.01.2019, 17:40
0.00 руб.
31.01.2019, 11:27
1
2
1
Здравствуйте! Прошу помощи в данном задании:
Пусть
и 
при 
Используя теорему 2.5 о пределе монотонной последовательности, докажите, что 
Пусть
и при Используя теорему 2.5 о пределе монотонной последовательности, докажите, что
Прикрепленные файлы:
Обсуждение
31.01.2019, 11:24
общий
31.01.2019, 12:08
Адресаты:
Сообщите, пожалуйста, какова формулировка теоремы 2.5, упомянутой Вами в задании?
Об авторе:
Facta loquuntur.
Facta loquuntur.
31.01.2019, 18:40
общий
это ответ
Здравствуйте, alina.poroshina124564!
По условию
Тогда из рекуррентной формулы получим, что для любого 
имеем 
то есть заданная последовательность ограничена снизу числом 
Методом математической индукции установим, что для любого
имеем 
то есть заданная последовательность ограничена сверху числом 
При этом
то есть заданная последовательность монотонно убывает. Будучи ограниченной при этом снизу и сверху, она, согласно теореме о признаке сходимости монотонной последовательности, имеет предел
Чтобы вычислить этот предел, воспользуемся его единственностью, откуда с учётом рекуррентной формулы получим
то есть
По условию
Тогда из рекуррентной формулы получим, что для любого имеем то есть заданная последовательность ограничена снизу числом Методом математической индукции установим, что для любого
имеем то есть заданная последовательность ограничена сверху числом При этомто есть заданная последовательность монотонно убывает. Будучи ограниченной при этом снизу и сверху, она, согласно теореме о признаке сходимости монотонной последовательности, имеет предел
Чтобы вычислить этот предел, воспользуемся его единственностью, откуда с учётом рекуррентной формулы получимто есть
5
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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