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- MHWLv 51 decade agoFavorite Answer
Observe that
ln (x + 1) / ln x = [ln (1 + 1/x) + ln x] / ln x = [ln (1 + 1/x) / ln x] + 1,
which clearly tends to 1 as x tends to infinity. Then, applying the ratio test to our given series:
[ (ln n+1)^1000 / 1.001^(n + 1) ] / [ (ln n)^1000 / 1.001^n ]
= (ln (n+1) / ln n)^1000 / 1.001
which tends to 1/1.001 < 1 as n tends to infinity, and so the series converges.
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