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Do you know how to get an asymptotic expansion up to $o(1)$ when $n\longrightarrow +\infty$ for $(u_n)_{n\in\mathbb{N}}$ defined by $u_0>1$ and $\forall n\in\mathbb{N},\; u_{n+1}=u_n+\ln(u_n)$ ? Thank you. Edit : if the problem was easier, I would like to find an equivalent for $v_{n+1}=v_n\times\ln(v_n)$.

P.Fazioli
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    Maybe What is a good asymptotic for $f_n = f_{n-1}+\ln(f_{n-1})$? ? – Sil Apr 29 '21 at 10:16
  • Thank you, I had not seen this topic. It seems that the expansion up to $o(1)$ is going to be difficult at least. I was expecting it to find asymptotic expansion for $v_{n+1}=v_n\times\ln(v_n)$. – P.Fazioli Apr 29 '21 at 10:25
  • @P.Fazioli I saw the recurrence $a_{n + 1} = p_{a_n }$ ($p_m$ is the $m$th prime) the other day on this site (https://math.stackexchange.com/q/4119019). Since $p_m \sim m\log m$, it is similar to you problem. The asymptotics of that sequence seems to be a rather difficult question as well. – Gary Apr 29 '21 at 11:24
  • Thank you for the reference. I am going to test numerical values to get an idea for an equivalent of $v_n$. – P.Fazioli Apr 29 '21 at 11:45
  • No, because the asymptotic expansion is not up to $o(1)$. The link does not allow to find an equivalent for the sequence $v_{n+1}=v_n\times\ln(v_n)$ when we take $\exp$. – P.Fazioli May 01 '21 at 09:33

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