# The surreal version of $e$

For a sequence $(x_{\alpha})$ of surreal numbers indexed by the set of all ordinal numbers, we say that $\lim x_{\alpha}=l$ ($l$ is a surreal number) if for each surreal $\epsilon>0$, there exists an ordinal $\beta$ such that $|x_{\alpha}-l|<\epsilon$ for each ordinal $\alpha>\beta$.

Consider the sequence $x_{\alpha}=(1+1/\alpha)^\alpha$, where for each surreal $x,y$, $x^y$ is defined by $\exp(y\log x)$.

Q1: What is the normal form of $x_{\omega}$? Is its real part equals to $e$?

Q2: What is $\lim x_{\alpha}$?

• What, $e$ is not surreal enough for you as it is? – Asaf Karagila May 30 '17 at 16:07
• The set of all ordinal numbers? – Nik Weaver May 30 '17 at 16:34
• I think you mean "standard part" rather than "real part". – Joel David Hamkins May 30 '17 at 19:46
• Of course many sequences indexed by the ordinal numbers do not converge at all in the surreals. That is, the left and/or right "set" has to be a proper class. Do you think this sequence is one? – Gerald Edgar May 30 '17 at 21:46
• @JoelDavidHamkins : Any surreal $x$ can be written in a unique way as the sum $x_{\prec 1} + x_{\asymp 1} + x_{\succ 1}$ where $x_{\prec 1}$, the infinitesimal part, is infinitesimal, $x_{\asymp 1}$, the real part, is real, and $x_{\succ 1}$, the purely infinite part, has only strictly positive exponents in its normal form, or equivalently is a logarithm of some $\omega^y$. – nombre May 30 '17 at 22:01

$\DeclareMathOperator{\ee}{e}$If $\varepsilon$ is an infinitesimal surreal, the quantity $\log(1+\varepsilon)$ is actually equal to the formal sum à la Hahn series $\sum \limits_{n \in \mathbb{N}} \frac{(-1)^n\varepsilon^{n+1}}{n+1}$, and $\log(1+\varepsilon) - \varepsilon$ is negligeable with respect to $\varepsilon$. So the proof for real numbers can be applied here to show that the sequence converges to $\ee$.
As for $x_{\omega}$, this is $\exp(1 - \frac{1}{2\omega} + \frac{1}{3\omega^2} - ...)$ where $\exp(- \frac{1}{2\omega} + \frac{1}{3\omega^2} - ...)$ is infinitesimally close to $1$ because $a:= -\frac{1}{2\omega} + \frac{1}{3\omega^2} - ...$ is an infinitesimal. So the real part of $x_{\omega}$ is $\ee$.
For now I don't see what its normal form because some combinatorial cleverness seems to be required in unfolding $\sum \limits_{n \in \mathbb{N}} \frac{a^n}{n!}$. I'll edit this answer if I find something. In the meantime you can try to find it too: we know that exponents of $\omega$ in the $a^n$ and $\exp(a)$ will be negative integers. It might be easier to compute the exponential sum directly if you can find relations between the coeffifients $q_{n,k}$ of $\omega^{-k}$ in $a^n$ for different values of $n$.
• Doesn't this argument show that the limit is actually e? It seems to show that $(1+\epsilon)^{1/\epsilon}$ is equal to $e$ to within $O(\epsilon)$, so if $\epsilon$ gets smaller than the inverse of any ordinal, it should be equal ot $e$ within the inverse of every ordinal. – Will Sawin May 30 '17 at 22:46
• @WillSawin : $\DeclareMathOperator{\ee}{e}$$\DeclareMathOperator{\Ord}{Ord}$Yes, this shows that the limit is $e$, and for that matter, that $\exp$ can be defined as $x \mapsto \lim \limits_{\alpha \in \Ord} (1 + \frac{x}{\alpha})^{\alpha}$. What's missing in the anwser is the normal form of $x_{\omega}$. – nombre May 31 '17 at 7:11
• So, were we to define exponentials from scratch, it would be enough to define $s^\alpha$, for $s$ a surreal and $\alpha$ an ordinal. Is it workable to introduce the exponential this way? Might it be simpler than the standard way? – Paolo Lipparini Jul 18 '17 at 22:29