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Консультация № 160907

19.02.2009, 16:44

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Образование

Помогите решить задачу №137 из сборника Демидовича . Условие такое :
Пусть числовая последовательность X1,X2,...,Xn,... удовлетворяет условию 0<=Xm+n<=Xm+Xn
(m,n = 1,2,3,...) Доказать , что lim(Xn/n) существует ( при n стремящемся к бесконечности )

Обсуждение

Неизвестный

19.02.2009, 17:51

общий

это ответ

Здравствуйте, Matemateg!
Из данного нам неравенства следует что 0[$8804$]x_n[$8804$]x₁+x₁+...+x₁=nx₁, 0[$8804$]x_n/n[$8804$]x₁, n=2,3,..., значит последовательность (x_n/n) ограничена и существует точная нижняя граница a=inf{x_n/n}. То есть для любого e>0 найдется такой номер m, что a[$8804$]x_m/m<a+e/2.
Теперь воспользуемся тем, что всякое целое число n можно представить как n=qm+r, где r одно из чисел 0,1,2,...,m-1.
x_n=x_qm+r[$8804$]x_m+x_m+...+x_m+x_r=qx_m+x_r,
x_n/n=x_qm+r/(qm+r)[$8804$](qx_m+x_r)/(qm+r)=(x_m/m)[$149$]qm/(qm+r)+x_r/n
a[$8804$]x_n/n<(a+e/2)[$149$]qm/(qm+r) + x_r/n<a+e/2+x_r/n.
Так как r у нас принимает значение от нуля и до m+1, то x_r ограничено и найдется такое N(e), что при n>N(e) 0[$8804$]x_r<e/2. А тогда a[$8804$]x_n/n<a+e/2+e/2=a+у при n>N(e) и lim (x_n/n)=a при n стремящимся к бесконечности.

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