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Консультация № 174366
19.11.2009, 12:16
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Разложить в ряд Маклорена функцию f(x). Указать область сходимости полученного ряда:
f(x)=arctgx/x
f(x)=arctgx/x
Обсуждение
21.11.2009, 03:27
общий
это ответ
Здравствуйте, Александров Анатолий Олегович.
Полагаю, что можно поступить следующим образом.
Воспользуемся тем, что
arctg x = x – x3/3 + x5/5 – x7/7 + … = Σn = 0∞ (-1)nx2n + 1/(2n + 1) (|x| < 1),
arctg x = ±π/2 – 1/x + 1/(3x3) – 1/(5x5) + 1/(7x7) – … = ±π/2 + Σn = 0∞ (-1)n + 1/((2n + 1)x2n + 1) (|x| > 1).
Тогда
(arctg x)/x = 1 – x2/3 + x4/5 – x6/7 + … = Σn = 0∞ (-1)nx2n/(2n + 1) (|x| < 1),
(arctg x)/x = ±π/2 – 1/x2 + 1/(3x4) – 1/(5x6) + 1/(7x8) – … = ±π/2 + Σn = 0∞ (-1)n + 1/((2n + 1)x2(n + 1)) (|x| > 1).
С уважением.
Полагаю, что можно поступить следующим образом.
Воспользуемся тем, что
arctg x = x – x3/3 + x5/5 – x7/7 + … = Σn = 0∞ (-1)nx2n + 1/(2n + 1) (|x| < 1),
arctg x = ±π/2 – 1/x + 1/(3x3) – 1/(5x5) + 1/(7x7) – … = ±π/2 + Σn = 0∞ (-1)n + 1/((2n + 1)x2n + 1) (|x| > 1).
Тогда
(arctg x)/x = 1 – x2/3 + x4/5 – x6/7 + … = Σn = 0∞ (-1)nx2n/(2n + 1) (|x| < 1),
(arctg x)/x = ±π/2 – 1/x2 + 1/(3x4) – 1/(5x6) + 1/(7x8) – … = ±π/2 + Σn = 0∞ (-1)n + 1/((2n + 1)x2(n + 1)) (|x| > 1).
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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