Bertrand series: Why is $\frac{1}{n^{\alpha}log^{\beta}(n)}$ not bounded for $\alpha < 0$

Asked Sep 19 '15 at 16:31

Active Feb 10 '18 at 11:41

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Why is $$\frac{1}{n^{\alpha}log^{\beta}(n)}$$ not bounded for $\alpha < 0$?

edited Feb 10 '18 at 11:41

asked Sep 19 '15 at 16:31

mavavilj

Becasue $n^{-\alpha}$ grows much faster to infinity for $\alpha<0$ than the logarithm does, (if it does at all, that is, if $\beta>0$; if $\beta \leq 0$ then you just have positive powers of sequences tending to infinity) – uniquesolution Sep 19 '15 at 16:35
@uniquesolution Why does it grow faster? – mavavilj Sep 19 '15 at 16:44
Here, read this. http://www.math.uconn.edu/~kconrad/blurbs/analysis/growth.pdf – uniquesolution Sep 19 '15 at 16:47
https://math.stackexchange.com/questions/2563446/on-convergence-of-bertrand-series-sum-limits-n-2-infty-frac1n-alpha – Guy Fsone Feb 10 '18 at 10:10

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