When deriving the exponential generating function of Bernoulli Numbers and its closed form, I've stumbled upon a huge, to my rigorous heart, problem - discontinuity of the closed form.
$$ B(t) = \sum_{n = 0}^{\infty} {B_n \frac{t^n}{n!}} = \frac{t}{e^t - 1} $$
In fact, the closed form has the singularity at $t = 0$. Doesn't it ruin the concept of generating functions?
As far as I know, a closed form of a gf is required to be analytic as a function. It is mainly because we need it to reproduce the encoded sequence by the formula: $$ B_n = B^{(n)}(0) $$
In our case, we cannot evaluate directly even $B_0$, as we have a zero denominator.
However, the singularity is removable. And if we put the closed form at $t = 0$ to be $1$, then it becomes analytic!
$$ \text{if $t = 0$ then } B(t) = 1 $$ $$ \text{otherwise } B(t) = \frac{t}{e^t - 1} $$
Is it reasonable to use closed form defined in such a way? Or am I mistaken somewhere?
Thank You in advance!