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Take $a_0=10^6$.

What is $a_n$ (asymptotically) where $a_{i+1}=a_i+\sqrt[\alpha]{a_i}$ where $\alpha>1$?

How fast does $a_n$ grow?

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    $\begingroup$ $a_n/n^2\to 1/4$, see my answer to essentially the same question here: math.stackexchange.com/questions/1485482/… $\endgroup$ – user138530 Oct 31 '15 at 2:40
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    $\begingroup$ That's for the square root; here we'll have something like $a_n\sim n^\frac{\alpha}{\alpha-1}$ $\endgroup$ – Ivan Neretin Oct 31 '15 at 12:17
  • $\begingroup$ @IvanNeretin can you show this below? $\endgroup$ – Turbo Oct 31 '15 at 12:58
  • $\begingroup$ Well, the main idea is the same as in Christian's answer to the other question, and the rest is just too boring. $\endgroup$ – Ivan Neretin Oct 31 '15 at 16:00

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