Compute: $\lim\limits_{n\to\infty} \frac{x_n}{\ln {n}}$

Let be the sequence $(x_n)_{n\geq0}$, $x_0$ a real number, and defined as follows: $$x_{n+1} = x_n + e^{-x_n}$$ Compute the limit:

$$\lim_{n\to\infty} \frac{x_n}{\ln {n}}$$

Luckily, I've found another post here with a very similar question. If you know other ways to solve it and want to share them then I'd be very grateful for that.

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Heuristically, this difference equation resembles the differential equation $x'(t) = e^{-x}$, which has solutions $x(t) = \ln(t + C)$, so I would expect the answer to be $1$.

In fact, let $x_n = \ln(n + s_n)$, and $y(t) = \ln(t+s_n)$. Then for $n \le t < n+1$ we have $y(t) > y(n) = x_n$, so \eqalign{ \ln(n+1+s_n) - \ln(n+s_n) &= \int_n^{n+1} \dfrac{dt}{t+s_n} = \int_n^{n+1} e^{-y(t)}\ dt\cr &< \int_n^{n+1} e^{-x_n}\ dt = e^{-x_n} = x_{n+1} - x_n\cr} i.e. $x_{n+1} - \ln(n+1+s_n) > x_n - \ln(n+s_n) = 0$, or $s_{n+1} > s_n$. Thus $x_n > \ln(n+s_0)$. But then $x_{n+1} - x_n = e^{-x_n} < \dfrac{1}{n + s_0}$ so $\displaystyle x_n < x_0 + \sum_{j=0}^{n-1} \frac{1}{j+s_0}$. Since both upper and lower bounds are $\ln(n) + O(1)$, we conclude that $x_n = \ln(n) + O(1)$.

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it seems that once you solve that differential equation everything is clear. Thanks! –  Chris's sis Aug 10 '12 at 8:24

We examine the asymptotic behavior of $(x_n)$ inch by inch.

Step 1. We first show that $x_n \to \infty$. Note that for any $x < 0$, there exists $y > 0$ such that $x + e^{-x} = y + e^{-y}$. Thus we may assume $x_0 \geq 0$. Then $(x_n)$ is monotone increasing, thus it tends to a limit $\ell \in [0, \infty]$. Taking limit to the recursive formula, we have $\ell = \ell + e^{-\ell}$, which is valid only if $\ell = \infty$.

Step 2. Let $y_n = e^{-x_n}$. Then $(y_n)$ decreases monotonely to $0$. Also, we have

$$y_{n+1} = \exp\left(-x_{n+1}\right) = \exp\left(-x_{n}-e^{-x_n}\right) = y_n e^{-y_n}.$$

Then we have

$$y_{n+1}^{-1} - y_n^{-1} = \frac{e^{y_n} - 1}{y_n} \xrightarrow[]{n\to\infty} 1.$$

Thus by Cesàro-Stolz theorem, we have

$$\lim_{n\to\infty} \frac{y_n^{-1}}{n} = 1.$$

This already shows that $\lim_{n\to\infty} \frac{x_n}{\log n} = 1$.

Step 3. Since

$$\frac{y_{n+1}^{-1} - y_n^{-1} - 1}{y_n} = \frac{e^{y_n} - 1 - y_n}{y_n^2} \xrightarrow[]{n\to\infty} \frac{1}{2},$$

we have

$$\frac{\left(y_{n+1}^{-1} - (n+1)\right) - \left(y_n^{-1} - n\right)}{\log(n+1) - \log n} \sim \frac{y_{n+1}^{-1} - y_n^{-1} - 1}{y_n} \xrightarrow[]{n\to\infty} \frac{1}{2},$$

where we used the fact here that $y_n \sim \frac{1}{n} \sim \log(n+1) - \log n$. Thus by Cesàro-Stolz theorem again, we have

$$\lim_{n\to\infty} \frac{y_n^{-1} - n}{\log n} = \frac{1}{2},$$

which in particular implies that

$$y_n = \frac{1}{n} + O\left(\frac{\log n}{n^2}\right).$$

Step 4. Finally, we have

$$\frac{y_{n+1}^{-1} - y_n^{-1} - 1 - \frac{1}{2}y_n}{y_n^2} = \frac{e^{y_n} - 1 - y_n - \frac{1}{2}y_n^2}{y_n^3} \xrightarrow[]{n\to\infty} \frac{1}{6}.$$

Let $s_n = y_1 + \cdots + y_n$. Since $y_n^2 \sim \frac{1}{n^2}$,

$$\frac{\left(y_{n+1}^{-1}-(n+1)-\frac{1}{2}s_n\right) - \left(y_n^{-1}-n-\frac{1}{2}s_{n-1}\right)}{\frac{1}{n^2}} \xrightarrow[]{n\to\infty} \frac{1}{6},$$

or simply

$$\left(y_{n+1}^{-1}-(n+1)-\frac{1}{2}s_n\right) - \left(y_n^{-1}-n-\frac{1}{2}s_{n-1}\right) = O\left(\frac{1}{n^2}\right).$$

Thus summing,

\begin{align*}y_n^{-1} &= n + \frac{1}{2}s_{n-1} + O(1) = n + \frac{1}{2}s_{n} + O(1) \\ &= n + \frac{1}{2}\sum_{k=1}^{n} \left( \frac{1}{k} + O\left(\frac{\log k}{k^2}\right)\right)+O(1) \\ &= n + \frac{1}{2}\log n + O(1), \end{align*}

which finally implies the asymptotic formula

$$y_n = \frac{1}{n} - \frac{1}{2}\frac{\log n}{n^2} + O\left(\frac{1}{n^2}\right),$$

or in terms of $x_n$,

$$x_n = \log n + \frac{1}{2}\frac{\log n}{n} + O\left(\frac{1}{n}\right).$$

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a very nice solution. –  Chris's sis Aug 10 '12 at 9:34
Having given this last result, it seems that another good question is to compute $\lim_{n\to\infty} ({x_{n}- \log n}) \frac{n}{\log n}$ :-) –  Chris's sis Aug 10 '12 at 15:14