# About a Dirichlet series [closed]

I would like to know if the following assertion is true: Let consider a real decreasing sequence $(t_n)$ of positive numbers with limit zero, if the series $\sum\limits_{n=1}^\infty(t_n)^a$ is divergent for all real $a$ in $[0,1[$ then the series $\sum\limits_{n=1}^\infty t_n$ is also divergent? thanks in advance

## closed as off-topic by Greg Martin, Jan-Christoph Schlage-Puchta, Lucia, fedja, Willie WongAug 31 '18 at 17:54

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Greg Martin, Lucia, fedja
If this question can be reworded to fit the rules in the help center, please edit the question.

• For some basic information about typing mathematical expressions on Stack Exchange sites, see: How does one type mathematical formulas on this site? To help you get started, I have edited your post - I hope this is at least approximately close to what you intended, but if not please edit your post further. It was a bit unclear whether you want to ask about sequence or about series - in case it helps you, you can write sum as \sum $\sum$ or \sum_{n=1}^\infty $\sum_{n=1}^\infty$. (To get math rendered, it has to be included between dollar signs.) – Martin Sleziak Aug 27 '18 at 9:55
• I mean " the series \sum tn^a is divergent for all a in [0,1[ , does it follow that the series \sum tn is also divergent"; thanks for your help – teller Aug 27 '18 at 10:01
• BTW I'd guess that (ca.classical-analysis-and-odes) would be a more suitable top-level tag than (nt.number-theory). But probably it's better if I leave this up to you - I do not want to go overboard with editing your question. – Martin Sleziak Aug 27 '18 at 11:39
• This isn't a Dirichlet series. – Greg Martin Aug 27 '18 at 18:59

What about $t_n=\frac{1}{n(\log n)^2}$?