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
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closed as offtopic by Greg Martin, JanChristoph SchlagePuchta, Lucia, fedja, Willie Wong Aug 31 '18 at 17:54
This question appears to be offtopic. 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
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What about $t_n=\frac{1}{n(\log n)^2}$?

1$\begingroup$ thanks a lot; i tried 2 days to prove it (without success), now i know more:its false $\endgroup$ – teller Aug 27 '18 at 11:20
\sum
$\sum$ or\sum_{n=1}^\infty
$\sum_{n=1}^\infty$. (To get math rendered, it has to be included between dollar signs.) $\endgroup$ – Martin Sleziak Aug 27 '18 at 9:55