The function $\sum_{0}^{\infty} x^n/n^n$

Question

The function $F(x) = \sum_{0}^{\infty} x^n/n^n$ may be familiar to many readers as an example sometimes used when teaching tests for absolute convergence of entire functions defined by power series. I know of no name for it, nor any use for it aside from pedagogical, so this is a pure curiosity question which I hope is acceptable.

The function seems to have one real zero around $x = -1.40376$; a single extremum, a minimum around $x = -5.71837$; and then to approach the $x$-axis from below asymptotically as $x$ goes to negative infinity. Is this true?

Answer 1

[Edited to outline the end of the argument that $f(-M) \rightarrow 0$ (and to correct a few typos etc. while I'm at it)]

Yes, $F(x) \rightarrow 0$ from below as $x \rightarrow -\infty$. The convergence is slow, and precise asymptotic analysis seems to be somewhat annoying because it involves the lower branch of the Lambert W function.

The massive cancellations in $\sum_{n=0}^\infty x^n/n^n$ for $x \rightarrow -\infty$ can be tamed by the familiar device of writing $$ \frac1{n^n} = \frac1{(n-1)!} \int_0^\infty t^{n-1} e^{-nt} dt $$ for $n=1,2,3,\ldots$. Multiplying by $x^n$, summing over $n>0$, and restoring the $n=0$ term $x^0/0^0=1$ yields $$ f(x) = 1 + x \int_0^\infty e^{txe^{-t}} e^{-t} dt. $$ Hence if $x=-M$ then $$ f(x) = f(-M) = 1 - M \int_0^\infty e^{-Mte^{-t}} e^{-t} dt, $$ and as $M \rightarrow +\infty$ the integral naturally splits into the parts $t \leq 1$ where $t e^{-t}$ is increasing and $t \geq 1$ where $t e^{-t}$ is decreasing. We let $u = t e^{-t}$, so the integrand becomes $e^{-Mu} du/(1-t)$. For $t<1$ we use Abel's power series $t = \sum_{m=1}^\infty m^{m-1} u^m/m!$ to expand the integral in an asymptotic series: $$ \int_0^1 e^{-Mte^{-t}} e^{-t} dt \sim \frac1M + \frac1{M^2} + \frac{2^2}{M^3} + \frac{3^3}{M^4} + \frac{4^4}{M^5} + \cdots $$ which is already enough to get $f(-M) < 0$ for large $M$. [Curiously the asymptotics of $\sum_{n=0}^\infty (-M)^n/n^n$ have led us to the divergent series $\sum_{n=1}^\infty n^n/M^n$.]

But the resulting bound $f(-M) < -1/M$ underestimates $|f(-M)|$: numerically $f(-100) \simeq -.1826$, $\phantom.$ $f(-1000) \simeq -.1180$, and $\phantom.$ $f(-10000) \simeq -.0899$, suggesting that $f(-M)$ decays only as $-1/\log M$ or so. The reason must be the $t>1$ part of the integral. On this part, $t = \log(1/u) + \log\log(1/u) + o(1)$ as $t \rightarrow \infty$, so the integral behaves to first order like $\int_0^{1/e} e^{-Mu} du / \log(1/u)$. Now $\log(1/u) \rightarrow 0$ as $u \rightarrow 0+$, but the convergence is slower than any positive power of $u$. Therefore, the integral is $o(1/M)$, which completes the argument that $f(x) \rightarrow 0$ as $x \rightarrow -\infty$; but the integral is not $O(1/M^\theta)$ for any $\theta > 1$, so $f(-M)$ decays slower than any positive power of $M$.

A more thorough asymptotic analysis of the $t>1$ integral as $M \rightarrow \infty$ looks routine but unpleasant, so I'll stop at this point; perhaps somebody else here will be interested in pursuing it further.

Answer 2

I wish to add a remark to Noam D. Elkies' beautiful answer. From the integral representation for $f$, putting $e^{-t}=s$ in the integral, $$f(-x)=1-x\int_0^\infty e^{-xte^{-t}} e^{-t}dt = 1-x\int_0^1 s^{sx}ds\, ,$$ so that, for $x\to \infty$, $ f(-x)=o(1)$ is equivalent to $$\int_0^1 xu(s)^xds=1+o(1)\, ,$$ where $u\in C([0,1])$ is the function $u(s):=s^s$. As a matter of fact, since $0\le u(s)\le 1$ for all $s$ and $u(s)=1$ only for $s=0$ or $s=1$, it turns out that the limit only depends on $u'(0)$ and $u'(1)$.

Since $u'(1)=1$, for any $\lambda < 1 < \mu$ there exists a $b < 1$ such that for all $s\in [b,1]$ there holds $$1+\mu(s-1) \le u(s)\le 1+\lambda(s-1)\, ,$$ so that $$x\big(1+\mu(s-1)\big)^x \le xu(s)^x\le x\big(1+\lambda(s-1)\big)^x\, .$$ Similarly, since $u'(0)=-\infty$, for any $\nu > 0$ there exists a $a > 0$ such that for all $s\in [0,a]$ $$u(s)\le1-\nu s\, ,$$ so $$xu(s)^x\le x\big( 1-\nu s\big) ^ x \, .$$

Moreover, since on any interval $[a,b]\subset\subset(0,1)$ the function $u$ is bounded away from $1$, it is clear that $\int_a^b xu(s)^xds=o(1)$ by uniform convergence to $0$.

Integrating over $s\in [ 0,1]$, and recalling that $\lambda < 1 < \mu$ and $\nu > 0$ were arbitrary, the inequalities above plainly give

$$\int_0^1 xu(s)^xds=\int_0^a x u(s)^xds+\int_a^b xu(s)^xds+\int_b^1 xu(s)^xds=1+o(1) \, ,$$ for $x\to \infty$.

Answer 3

There is a paper of G. H. Hardy, where this function is studied in great detail:

G. H. Hardy, On the integral function $ \Phi_{ a,\alpha,\beta}(z)=\sum x^n/(n+a)^{\alpha n+\beta}$, Quarterly J. Math., 5 (1906) 37, 369-378. (Collected papers of G. H.Hardy, vol. IV, p. 128).

Answer 4

I worked out the ideas of Noam Elkies.

A transformation of the integration variable $t$ to $\tau:=t e^{-t}$ gives for $0 \leq t \leq 1$ and $ 1 \leq t < \infty$,

$$ t = t_1(\tau) := - W_0(-\tau) \mathrm{\ \ for \ 0 \leq t \leq 1} \\ t = t_2(\tau) := - W_{-1}(-\tau) \mathrm{\ \ for \ 1 \leq t < \infty}, $$ where $W_0$ and $W_{-1}$ are the principal and "lower" branches of the Lambert-$W$ function, respectively, see the pictures for further illustration.

The picture below shows $t e^{-t}$ with the branch that maps to $-W_{0}(-\tau)$ ($-W_{-1}(-\tau)$) shown as solid (dashed) line.

The functions $t_{1}(\tau)$ (solid line) and $t_{2}(\tau)$ (dashed line) are shown below.

The transformation of the integration variable gives $$ d t= \frac{-W_0(-\tau)}{\tau (1 + W_0(-\tau))} d \tau \mathrm{\ \ for \ 0 \leq t \leq 1} \\ d t = \frac{-W_{-1}(-\tau)}{\tau (1 + W_{-1}(-\tau))} d \tau \mathrm{\ \ for \ 1 \leq t < \infty}, $$ and further $$ \frac{-W_0(-\tau)}{\tau (1 + W_0(-\tau))} \exp\left( W_0(-\tau)\right) = \frac{1}{1 + W_0(-\tau)} \\ \frac{-W_{-1}(-\tau)}{\tau (1 + W_{-1}(-\tau))} \exp\left( W_{-1}(-\tau)\right) = \frac{1}{1 + W_{-1}(-\tau)} $$ The integration intervals map as $$ t\in [0,1] \rightarrow \tau\in [0, e^{-1}] \\ t\in [1,\infty] \rightarrow \tau\in [e^{-1}, 0] $$ The function can then be written as $$ f(-x)=1-x \int_{0}^{\infty} d t \ e^{-t}\ e^{-x t e^{-t}} \\ = 1-x\int_{0}^{e^{-1}} d \tau \ \left( \frac{1}{1 + W_{0}(-\tau)} - \frac{1}{1 + W_{-1}(-\tau)} \right) e^{- x \ \tau} $$ The Lambert functions can now be expanded for small arguments using $$ W_{0}(-\tau)\sim 1-\tau \\ W_{-1}(- e^{-\eta}) \sim -\eta - \ln \eta - \frac{\ln \eta}{\eta} + \frac{\ln \eta \ (\ln \eta - 2)}{2\ \eta^2}. $$ The resulting integrals can be evaluated to give $$ f(-x) = -\frac{1}{\ln x} + \frac{\gamma - 1 + \ln(\ln(x))}{(\ln x)^2}(1 + O(e^{-x/e})) + O((\ln x)^{-3}), $$ with Euler's $\gamma$. Higher order asymptotic expansions follow from inclusion of higher order asymptotics of the Lambert functions.