级数求和函数,求解答
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利用绝对收敛数列:1 + x + x^2 + ... + x^n, n from 0 to oo = 1/(1-x), |x| < 1
两边同时积分from 0 to x, x + x^2/2 + x^3/3 + ... + x^(n+1)/(n+1) = - ln(1-x)
两边同除 x, 1 + x/2 + x^2/3 + ... + x^n/(n+1) = - (1/x) ln(1-x)
原级数的和 = x/2 + x^2/3 + ... + x^n/(n+1) = - (1/x) ln(1-x) - 1
因为边界点 x = 1处发散,x = -1处条件收敛,其收敛域:[-1, 1).
两边同时积分from 0 to x, x + x^2/2 + x^3/3 + ... + x^(n+1)/(n+1) = - ln(1-x)
两边同除 x, 1 + x/2 + x^2/3 + ... + x^n/(n+1) = - (1/x) ln(1-x)
原级数的和 = x/2 + x^2/3 + ... + x^n/(n+1) = - (1/x) ln(1-x) - 1
因为边界点 x = 1处发散,x = -1处条件收敛,其收敛域:[-1, 1).
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