Friday, March 23, 2018

sequences and series - Comparison test suminftyk=1frac1+ln(k)k



k=11+ln(k)k



Determine whether it converges or diverges.




I don't think I could do limit comparison test because the ln(k) messed me up. Pretty sure I could do this with integral test but I think this is possible with comparison test as well. Could someone tell me if it is? For instance I'm looking for a bk value that is 0akbk



My textbook uses bk=1k, but how is the hypothesis met with this? 1+ln(k)k1k



Thats wrong it should be akbk n1


Answer



The book is correct. Note that 1+log(k)1 for all k1. Hence,



1+log(k)k1k




Since the harmonic series diverges, then the series k=11+log(k)k diverges by comparison. That is to say, the series of interest dominates the divergence harmonic series.


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