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If the integral test applies, use it to determine whether the series converges or diverges.  ∑ 1, ∞   (ln(n))2/n

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Andre W. | Friendly tutor for ALL math and physics coursesFriendly tutor for ALL math and physics ...
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Since f(x)=(ln(x))²/x is positive and continuous on [1,∞), the integral test is applicable to the series
n=1  (ln(n))²/n.
Consider the improper integral
1(ln(x))²/x dx = limb→∞1 b (ln(x))²/x dx
= limb→∞ [(ln(x))³/3]b1
which diverges, so the series ∑ n=1  (ln(n))²/n also diverges.
Richard P. | Fairfax County Tutor for HS Math and ScienceFairfax County Tutor for HS Math and Sci...
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It is clear that this series diverges because, apart from the first two terms, each term is greater than 1/n. 
The series  ∑ 1,∞ 1/n   diverges.