Can we compute the infinite series of the cdf of a standard normal distribution $\sum^{\infty}_{n=0} \Phi(-\sqrt{n})$?

Question

I've tried plugging the series in Wolfram to first of all check if it converges but it doesn't return anything useful. Moreover, just computing $\Phi(-100)$ returns something useless. Looking at the distribution function of the standard normal distribution it is obvious that $\Phi(-100) \approx 0$, but I guess the same can be said about $1/100$, while the infinite series of $1/n$ is infinity.

Is this series even defined, i.e. can it be computed, or are some problems that don't allow computation?

Answer 1

$\Phi(-t) \sim \frac{e^{-t^2/2}}{\sqrt{2\pi} t}$ as $t \to \infty$, so your series does converge. According to Maple, the sum is approximately $$ .83116074272973488211268081565524436110357388769594$$ I doubt that there is a closed form expression for this.