设f(x)在区间[a,b]上满足a≤f(x)≤b,且有|f’(x)|≤q<1,令un=f(un-1)(n=1,2,…),u0∈[a,b],证明:级数绝对收敛.

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问题 设f(x)在区间[a,b]上满足a≤f(x)≤b,且有|f’(x)|≤q<1,令un=f(un-1)(n=1,2,…),u0∈[a,b],证明:级数绝对收敛.

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答案由|un+1-un|=|f(un)-f(un-1)|=|f’(ξ1)| |un-un-1|≤q|un-un-1|≤q2|un-1-un-2|≤…≤qn|u1-u0|且[*]绝对收敛.

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