Thursday, August 3, 2017

Alternating series: sumlimitsinftyn=1(1)n1fracln(n)n convergence?



Determine whether the series converges absolutely, conditionally or diverges?



n=1(1)n1ln(n)n




I know that |an| diverges by using the comparison test:



ln(n)n>1n

and the smaller, r.h.s being the divergent harmonic series.



So, should my conclusion for the alternating series be divergent or convergent conditionally*?






* How to estimate whether the alternating series terms are cancelling?


Answer




Let f(x)=lnxx so f(x)=1lnxx20 for xe and so the sequence (lnnn)n3 is decreasing to 0. Apply now the alternating series criteria to conclude the convergence of the series.


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