Proving that a certain sequence is bounded from above

Question

Let $p_1,p_2,p_3,..$ be the sequence of primes in increasing order ($p_1=2,p_2=3,...$) .Let $x_n$ be given by:

$$x_n=\frac{1}{p_1}+\frac{1}{p_2}+...+\frac{1}{p_n}-\sum_{i=1}^{n-1}\sum_{j=i+1}^n\frac{1}{p_ip_j}$$

Question: Is it true that $x_n<1$ for every $n\geq 1$ ?

Note: I haven't written a program to check my conjecture for large values of $n$ (since I don't have a suitable software to do so). I only checked it for small $n$ manually. It would be great if someone can check the conjecture first for large values of $n$.

Thank you

Answer 1

Looking at

$$x_{n+1} - x_n = \frac{1}{p_{n+1}} - \sum_{i=1}^n \frac{1}{p_ip_{n+1}} = \frac{1}{p_{n+1}}\left(1 - \sum_{i=1}^n \frac{1}{p_i}\right),$$

we see that $x_{n+1} < x_n$ for $n \geqslant 3$, since $\frac{1}{2} + \frac{1}{3} + \frac{1}{5} = \frac{31}{30} > 1$.

Hence the maximum is

$$x_3 = \frac{1}{2} + \frac{1}{3} + \frac{1}{5} - \frac{1}{6} - \frac{1}{10} - \frac{1}{15} = \frac{21}{30} < 1.$$