I was considering the following problem:
Somebody uses a backup for something (e.g. backups a file) and the backup is equally reliable as original storage. The storage is not perfectly reliable and it will eventually fail. Storage is being checked at regular intervals if it has failed and there is a small possibility $p$ that one of them will fail at the given time interval. The probability that they will both fail at the same interval is equal to the probability that one will fail during the interval when another fails, so this probability is also $p$. Probability that one intercepts one failure before the other (and therefore has any chance to act upon it), the "probability that backup worked" is therefore:
\begin{equation} P_1(p)=(1-p) \end{equation}
One could use the simplest strategy to re-backup once one storage failed, but then the "probability that backup worked" would approach zero and one is certain to eventually lose what was backed up.
Instead, after one storage fails, one uses two backups. The probability that all three storages fail in the same interval is $p^2$, and "probability that backup worked" is previous such probability times that they didn't fail at the same interval:
\begin{equation} P_2(p)=(1-p)(1-p^2) \end{equation}
Repeating the procedure, one later uses three backups, and increases number each time if the backup worked, so in $n$-th iteration, "probability that backup worked" is:
\begin{equation} P_n(p)=(1-p)(1-p^2)\cdot \ldots \cdot (1-p^n) \end{equation}
Continuing indefinitely, "probability that backup worked" is ultimately:
\begin{equation} P_\infty(p)=\prod_{k=1}^\infty (1-p^k) \end{equation}
Which brings me to the question: Is there any closed form for this expression (including special functions)? Getting the first terms in the Taylor expansion is relatively straightforward, but I fail to notice any pattern in it to be able to write it in a closed form.
Also, finding some differential equation for which the function \begin{equation} f_\infty(x)=\prod_{k=1}^\infty (1-x^k) \end{equation} is a solution would also help.
I would like to analyze this function and see the its graph.
Euler showed that $$ \prod_{k=1}^\infty(1-x^k) = \sum_{k=-\infty}^\infty (-1)^kx^{k(3k-1)/2}; $$ the result is known as as the pentagonal number theorem. The sum should converge when $0\leq x=p<1$.