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证明: $dps{vlm{n}sed{sum_{k=2}^n frac{1}{kln k}-lnln n}}$ 存在 (有限).
证明: 设 $dps{a_n=sum_{k=2}^n frac{1}{kln k}-lnln n}$, 则 $$eex ea a_{n+1}-a_n&=frac{1}{(n+1)ln(n+1)}-lnln(n+1)+lnln n\ &leq int_n^{n+1}frac{1}{xln x} d x-lnln(n+1)+ln ln n\ &=0,\ a_n&=sum_{k=2}^n frac{1}{kln k}-lnln n\ &geq sum_{k=2}^n int_k^{k+1}frac{1}{xln x} d x -lnln n\ &=int_2^{n+1} frac{1}{xln x} d x-lnln n\ &=lnln(n+1)-lnln 2-lnln n\ &geq -lnln 2. eea eeex$$ 由单调有界定理即知结论成立.