Error of prime guess So, I wanted to investigate what the probability of a randomly selected number to be prime was, i.e.
If I check if it's even, it'd be $1-1/2 = 1/2$, so $50:50$
Checking with 3, it'd be $1-1/2-1/6 ≈ 0.33$ since we've already checked all the even multiples of 3 
Checking five we get $1-1/2-1/6-1/30$ with the same reasoning.
So the just the series $1/2+1/6+1/30+1/210 + \cdots$ or the sum of $1/p!$, Where $p!$ is supposed to represent the product all primes multiplied up to the prime $p$ . I know that it converges to som value since the series $1/n!$ approaches $e$. And this one grows even faster. 
My question is, will it approach 1? 
Logically, yes it would since if we go on forever, all primes are accounted for and thus $1$ minus the series should give a error margin of $0$. But checking with the first $25$ primes, it's grows slower than Java will show me more decimals. It seems to converge to $0.705$, double check if you want to. It may very well be like the harmonic series (that is, it converges really slow), but I've already showed that it converges, so is there a way to prove that it approaches $1$?
Thank in advance
 A: The error is already in the equation for the fraction of numbers that are not divisible by $2$, $3$, $5$ etc. This is not
$$
1-\sum_{i=1}^{N} \frac{1}{\prod_{j=1}^i p_j}
$$
It must be 
$$
\frac{\phi\left(\prod_{i=1}^N p_i\right)}{\prod_{i=1}^N p_i} = \prod_{i=1}^{N} \left( 1-\frac{1}{p_i}\right)
$$
instead ($\phi$ is Euler's totient function), because you are actually looking for is the number of integers between $1$ and $\prod p_i$ that are relatively prime to $\prod p_i$. And the fraction of numbers that are divisible by none of the primes up to $p_N$ can be simplified to the formula given above.
It can be shown that this expression diverges to $0$ for $N\rightarrow\infty.$ (An infinite product is actually said to diverge if the partial products converge to $0$.) This follows from the convergence criteria of inifinite products and from the fact that the sum of the reciprocals of the primes diverges.
We can also use an example. Look at the first $3$ primes, $2$, $3$ and $5$. The pattern of numbers that are not divisible by one of those primes repeats every $30$ numbers, they are $30k+a$ with $a\in\{1,7,11,13,17,19,23,29\}$. This is a probability of $8/30$ while your formula would give a probability of $9/30$.
