How to find the limit for the quotient of the least number $K_n$ such that the partial sum of the harmonic series $\geq n$ Let $$S_n=1+1/2+\cdots+1/n.$$ Denote by $K_n$ the least subscript $k$ such that $S_k\geq n$. Find the limit $$\lim_{n\to\infty}\frac{K_{n+1}}{K_n}\quad ?$$

We know that $H_n=\ln n + \gamma +\epsilon(n)$, where $\epsilon(n)\approx \frac{1}{2n}$ and in any case $\epsilon(n)\rightarrow 0$ as $n\rightarrow \infty$.  If $m=H_n$ we may as a first approximation solve as $n=e^{m-\gamma}$.  Hence the desired limit is $$\lim_{m\rightarrow \infty} \frac{e^{m+1-\gamma}}{e^{m-\gamma}}=e$$
For a second approximation, $m=\gamma + \ln n +\frac{1}{2n}=\gamma+\ln n+\ln e^{\frac{1}{2n}}=\gamma+\ln ne^{\frac{1}{2n}}$.  This may be rearranged as $ne^{\frac{1}{2n}}=e^{m-\gamma}$.  This has solution $$n=-\frac{1}{2W(-e^{\gamma-m}/2)}$$
where $W$ is the Lambert function.  Hence the desired limit is now $$\lim_{m\rightarrow \infty}\frac{W(-e^{\gamma-m}/2)}{W(-e^{\gamma-m-1}/2)}=e$$
Although not a proof, this is compelling enough that I'm not going to think about the next error term.
