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Задать вопрос экспертам

Консультация № 185014

29.12.2011, 22:13

78.20 руб.

29.12.2011, 23:47

0 1 1

Образование

Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
1) Вычислить определённый интеграл
1
?(sinx^2)/(x^2)dx
0
с абсолютной погрешностью до 0.001, разложив подынтегральную функцию в степенной ряд и проинтегрировав его почленно.

2) исследовать сходимость числового ряда
?
? (5^r)/((3^r)(2*r+1))
r-1

Спасибо!!!

Обсуждение

давно

Гордиенко Андрей Владимирович

Мастер-Эксперт
17387
18345

29.12.2011, 23:07

общий

это ответ

Здравствуйте, Vikka!

Рассмотрим первое задание. Воспользуемся известным разложением
sin u = u - u³/3! + u⁵/5! - u⁷/7! + ... (|u| < [$8734$]).
Положив u = x², получим
sin x² = x² - x⁶/3! + x¹⁰/5! - x¹⁴/7! + ...,
(sin x²)/x² = 1 - x⁴/3! + x⁸/5! - x¹²/7! + ... .

Тогда
₀[$8747$]¹(sin x²)dx/x² [$8776$] ₀[$8747$]¹(1 - x⁴/3! + x⁸/5! - x¹²/7! + ...)dx = (x - x⁵/(3!5) + x⁹/(5!9) - x¹³/(7!13) + ...)|₀¹ =
= 1 - 1/30 + 1/1080 - 1/65520 + ... [$8776$] 1 - 0,0333 + 0,0009 [$8776$] 0,9676 [$8776$] 0,968.

Рассмотрим ряд [$8721$]_{r = 1}^[$8734$] 5^r/(3^r(2r + 1)). Имеем
a_r = 5^r/(3^r(2r + 1)), a_{r + 1} = 5^{r + 1}/(3^{r + 1}(2r + 3)).

Воспользовавшись признаком Даламбера, находим предел отношения a_{r + 1}/a_r при r [$8594$] [$8734$]:
a_{r + 1}/a_r = 5^{r + 1}/(3^{r + 1}(2r + 3)) : 5^r/(3^r(2r + 1)) = (5/3)(2r + 1)/(2r + 3) = (5/3)(2 + 1/r)/(2 + 3/r) = 5/3 > 1,
следовательно, заданный ряд расходится.

С уважением.

Об авторе:
Facta loquuntur.

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