$\newcommand{\logit}{\operatorname{logit}}$
A series may "diverge to $\infty$" or "diverge to $-\infty$"; a product may "diverge to $\infty$" or "diverge to $0$".
postscript in response to comments:
If $\lim_{n\to\infty}\prod_{i=1}^n a_i = 0$, it is standard to say that the product $\prod_{i=1}^\infty a_i$ "diverges to $0$". The product diverges to $0$ iff the sum $\sum_{i=1}^\infty\log a_i$ diverges to $-\infty$.
Why do we not say of a series that it "converges to $+\infty$" rather than that it "diverges to $+\infty$"? The object $+\infty$ is an absorbing element with respect to addition: $+\infty+x=+\infty$ for $x\in\mathbb R$. If we don't allow this kind of arithmetic, we still say that if $\sum_{i=1}^\infty b_i$ diverges to $+\infty$, then $x+\sum_{i=1}^\infty b_i$ diverges to $+\infty$, for $x\in\mathbb R$. The same applies to all other instances in this question: If $\prod_{i=1}^\infty a_i$ diverges to $0$, then $x\prod_{i=1}^\infty a_i$ diverges to $0$ for every $x\in\mathbb R^+$. Similarly in the example below, once the conditional probability given the data is $1$, adding a finite amount of additional information leaves it at $1$.
end of postscript
Now suppose we have either of two coins $A$ and $B$ and tossing $A$ gives $H$ more often than tossing $B$. Then $$ \begin{align} \logit \Pr(A\mid\text{outcome of first $n$ tosses}) & = \log\frac{\Pr(A \mid \text{outcome of first $n$ tosses)}}{\Pr(B \mid \text{outcome of first $n$ tosses})} \\[10pt] & = \logit \Pr(A) + \log\frac{\Pr(\text{outcome}\mid A)}{\Pr(\text{outcome}\mid B)}. \end{align} $$ If we get $H$ every time, then $\logit\Pr(A\mid\text{outcome})\to\infty$ as $n\to\infty$ and if we get $T$ every time, then $\logit\Pr(A\mid\text{outcome})\to-\infty$ as $n\to\infty$.
Clearly it makes sense to say that in the former case $\Pr(A\mid\text{outcome})$ "diverges to $1$", since you're approaching an absorbing boundary. By "absorbing", I mean that if you were to actually reach $0$ or $1$, you would be stuck there and could not get away from it.
My question is simply whether this phrase "diverges to $1$" occurs in that strange place called Reality, in particular in published books, journal articles, and the like.
Usage note: It appears that some non-statisticians may be unfamiliar with the term "logit". The first syllable is pronounced with a long "o" as in "low" and the "g" is like the "j" in "jet". It's defined as $$ \logit p = \log \frac{p}{1-p}. $$ Clearly $\logit p\to+\infty$ as $p\uparrow 1$ and $\logit p \to-\infty$ as $p\downarrow 0$, and $\logit(1/2) = 0$, and $\logit(p) = -\logit(1-p)$.
