Let $X$ be the random variable obtained as the maximum of a throw of $m$ dice (each of which is $n$-sided). In other words, $X = \max\{l_1,\cdots, l_m\}$ where $l_i$ can take any value between $1$ and $n$. Using the relationship between the sum of consecutive powers and Bernoulli numbers, one obtains the following formula for the probability of $X=k$:

$$Pr(X=k) = \frac{1}{n^m}[k^m -\frac{1}{m}B_m(k) + \frac{1}{m}B_m)]$$

where $B_m$ is the $m$th Bernoulli number and $B_m(k)$ is the $m$th Bernoulli polynomial evaluated at $k$. Clearly, if we let $n$ approach infinity, then the quantity above will approach zero. However, I am unsure what happens if we instead let $m$ approach infinity. The question then is does the following limit exist

$$Pr_{\infty}(X=k) := \lim_{m\to\infty}(\frac{1}{n^m}[k^m -\frac{1}{m}B_m(k) + \frac{1}{m}B_m)]) $$