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Консультация № 177346
19.03.2010, 11:09
42.39 руб.
0
2
2
Добрый день. Нужна помощь.
Условие: решить обратную задачу нахождения функции по ее ряду Маклорена
ряд: S(x) = 1 + x/2 + ... + x^(n-1)/n +...
На сколько я понимаю, нужно проинтегрировать ряд, затем найти сумму получившегося ряда, сумму продифференцировать.
Интегрируем: S1(x) = x + x^2/4 + ... + x^n/(n^2) +...
Как найти сумму этого ряда?
Условие: решить обратную задачу нахождения функции по ее ряду Маклорена
ряд: S(x) = 1 + x/2 + ... + x^(n-1)/n +...
На сколько я понимаю, нужно проинтегрировать ряд, затем найти сумму получившегося ряда, сумму продифференцировать.
Интегрируем: S1(x) = x + x^2/4 + ... + x^n/(n^2) +...
Как найти сумму этого ряда?
Обсуждение
Неизвестный
19.03.2010, 11:57
общий
это ответ
Здравствуйте, Лубошев Е.М..
Может быть, Вам поможет следующее:
Известно разложение в ряд функции ln(1+x) на интервале [-1, 1): ln(2+x)=x-x^2/2+..+(-1)^(n+1)*x^n/n+... из чего получается:
ln(1-x)=-x-x^2/2+..+(-1)^(n+1)*(-1)^n*x^n/n+...=-x*(1+x/2+...+x^(n-1)/n)+...)=-x*S(x), S(x)- нужная Вам сумма ряда.
Тогда S(x)=-ln(1-x)/x (на интервале (-1, 0) U(0,1] , при x=0 S=1)
Может быть, Вам поможет следующее:
Известно разложение в ряд функции ln(1+x) на интервале [-1, 1): ln(2+x)=x-x^2/2+..+(-1)^(n+1)*x^n/n+... из чего получается:
ln(1-x)=-x-x^2/2+..+(-1)^(n+1)*(-1)^n*x^n/n+...=-x*(1+x/2+...+x^(n-1)/n)+...)=-x*S(x), S(x)- нужная Вам сумма ряда.
Тогда S(x)=-ln(1-x)/x (на интервале (-1, 0) U(0,1] , при x=0 S=1)
Форма ответа
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