Reading through a real analysis text book I noticed that [itex]\sum[/itex] 1/K(adsbygoogle = window.adsbygoogle || []).push({});

diverges but [itex]\sum[/itex] 1/K^{1+[itex]\epsilon[/itex]}converges for all [itex]\epsilon[/itex] > 0.

This is confusing because 1/k will eventually be equally as small as the terms in 1/K^{1+[itex]\epsilon[/itex]}and therefore it should also converge. It may take much longer but surely the numbers eventually become equally small and adding more on becomes pointless.

I also saw that one can prove this using the integral test but the integral test isn't the 'decider' there must a more rigorous reason.

Essentially what I'm asking is who or what mathematical concept determined the required rate for convergence because it seems arbitrary to me.

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# How fast does convergence have to be

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