Chapter 8: Infinite Sequences and Series
Section 8.3: Convergence Tests
Determine if the series ∑n=2∞lnnn diverges, converges absolutely, or converges conditionally.
If it converges conditionally, determine if it also converge absolutely.
Although an=lnn/n→0 as n→∞, the divergence of the given series can be shown by both the Limit-Comparison and Integral tests (Table 8.3.1).
Since ddxlnxx=1−lnxx2, which is negative for x>e≐2.7, the sequence lnn/n is eventually monotone decreasing to zero, so the Integral test applies, and gives
∫3∞lnxx ⅆx = ∞
Since this integral diverges, so too does the given series.
Alternatively, since limn→∞ lnn/n1/n=limn→∞lnn = ∞, by part (3) of the Limit-Comparison test the given series diverges because the comparison series is the divergent harmonic series.
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