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Уважаемые эксперты! Пожалуйста, ответьте на вопрос: Как доказать следующую теорему: Пусть P - произвольное натуральное число, P>1. Существует и единственно представление этого числа в виде степенного ряда(аддитивно-мультипликативная запись) X = An*P^n + An-1*P^n-1+...+ A1P + a0, где 0<=Ai<P, 0<=i<=n, n не равно 0? Доказывается два утверждения: существование и единственность представления.
Тогда я выложу само доказательство и укажу на те моменты, которые мне непонятны. Во-первых, как графически можно изобразить деление интервала [P^n;P^n+1) на P-1 равную часть? Ведь P-1 - это последняя цифра в алфавите, скажем, 9. Были мысли, что числа из семейства P - это любые числа,но так как они указаны в степенном ряду, они могут быть в начальной числовой последовательности только базисом. Можно подставить в интервал число основания(10). Получим [10^1;10^2) - длина полуинтервала равна .разности = 89. при P принадлежит натуральному множеству N. Но если мы разделим остальные такие полуинтервалы, то разность будет другая, а значит, что они не равны. Может, если это прояснится, то будет прозрачно всё остальное доказательство. Далее, индексы полуинтервала [xk; xk+1] не могут быть больше P-1, потому что P-1 - последнее число в ряду k = 1,2,...P-1. Тогда индексы при x не могут быть больше 9, а только равны и меньше. Из этого же получаем, что полученный интервал [xk;xk+1] состоит из 9 частей. Почему в таком случае? Пока так
Ответ на вопрос консультации я дал. Если Вы имели в виду тонкости доказательства, с которым Вы уже были знакомы, то нужно было соответственно сформулировать и вопрос, который Вы задаёте уже второй раз в мини-форумах консультаций...
Начните с первого предложения, которое Вам непонятно. Пожалуйста, будьте кратки и конкретны: тяжело читать многословные сообщения с ускользающим смыслом.
Во втором доказательстве, которое вы привели, я испытываю трудности с понимаем сути индексов в неравенствах. Например, там, где я обвёл в красную рамку, я не понимаю, как следует понимать ak < P-1 и ai< P-1? Это означает два разных одночлена одного степенного ряда? По аналогии они должны быть с одним коэффициентом, ведь рассматриваются случаи a0 равно и меньше P - 1.
Да, натурального числа X. Как представить графически, как разделили полуинтервал [P^k;P^k+1) на P-1 равную часть? Предлагаю для удобства использовать десятичную систему счисления. P-1 здесь - это 9. Разделили каждый интервал [P^k;P^k+1), т.е. [10^0;10^1), [10^1;10^2),...[10^k;10^k+1). Тогда проблема в том, что длина не каждого такого интервала делится нацело на 9. Учитывая, что 9,90,900 мы не включаем, получим, что длины будут равны: 8,899,8999. Пока такие мысли.
Ответьте, пожалуйста, сначала на мой первый вопрос, и мы разберёмся с ним. Затем ответьте, пожалуйста, на мой второй вопрос, и мы разберёмся с ним. Потом разберёмся с Вашим вопросом ко мне. И так далее, иначе обсуждение превратится в два несвязанных между собой монолога.
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Facta loquuntur.
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