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Консультация № 170516
17.07.2009, 17:38
0.00 руб.
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Пользуясь разложением функции cosx2 в степенной ряд,вычислить приближённое значение этой функции при x=Пи/6 с точностью до 10-5.
Обсуждение
17.07.2009, 18:33
общий
это ответ
Здравствуйте, Alik4546.
Разложение функции cos u в ряд по степеням u имеет вид
cos u = Σ n = 0∞ (-1)n ∙ u2n/(2n)! = 1 – u2/2! + u4/4! - … + (-1)n ∙ u2n/(2n)! + … .
Положим u = x2 = (π/6)2 = π2/36 ≈ 0,274156. Поскольку
u0 = 1,
u1 = -(π2/36)2/2 = -π4/2592 ≈ -0,037581,
u2 = (π2/36)4/24 = π8/40310784 ≈ 0,000235,
u3 = -(π2/36)6/720 = -π12/1567283281920 ≈ -5,9 ∙ 10-7, |u3| < 10-5,
для решения поставленной задачи достаточно первых трех слагаемых:
cos (π/6)2 ≈ 1 - 0,037581 + 0,000235 ≈ 0,96265.
Ответ: 0,96265.
С уважением.
Разложение функции cos u в ряд по степеням u имеет вид
cos u = Σ n = 0∞ (-1)n ∙ u2n/(2n)! = 1 – u2/2! + u4/4! - … + (-1)n ∙ u2n/(2n)! + … .
Положим u = x2 = (π/6)2 = π2/36 ≈ 0,274156. Поскольку
u0 = 1,
u1 = -(π2/36)2/2 = -π4/2592 ≈ -0,037581,
u2 = (π2/36)4/24 = π8/40310784 ≈ 0,000235,
u3 = -(π2/36)6/720 = -π12/1567283281920 ≈ -5,9 ∙ 10-7, |u3| < 10-5,
для решения поставленной задачи достаточно первых трех слагаемых:
cos (π/6)2 ≈ 1 - 0,037581 + 0,000235 ≈ 0,96265.
Ответ: 0,96265.
С уважением.
5
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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