Does this limit with a Product Exist?

Question

I have been working on $$n \cdot \left(1-\prod_{i=1}^n (1-\frac{1}{p_i})\right)$$ and would like to know whether

$\lim \limits_{n \to \infty} (n \cdot \left(1- \prod_{i=1}^n (1-\frac{1}{p_i})\right))$

exists.

Answer 1

$$n-n \cdot \left(\prod_{i=1}^n (1-\frac{1}{p_i})\right) \geq n-n \cdot \left((1-\frac{1}{p_1})\right) \to \infty$$

Answer 2

N.S has already given a pretty direct solution. A slight twist on that which uses a nice little result about prime is that you can call in the divergence of the Harmonic Series $ H_{N}$.

$$\lim_{N \to \infty}H_{N} = \sum_{n=1}^{\infty}\frac{1}{n} = \prod_{\textrm{primes}}\frac{1}{1 - \frac{1}{p}}$$

Flipping the fraction upside down we get, abusing notation/limits,

$$ \prod_{\textrm{primes}}\left(1 - \frac{1}{p}\right) = \frac{1}{H_{\infty}}$$

Since the Harmonic series diverges it follows that $\lim_{n\to\infty} \prod_{i=1}^{n}\left(1 - \frac{1}{p_{i}}\right) = 0$ and so the ratio of the two terms $n$ and $n\prod_{i=1}^{n}\left(1 - \frac{1}{p_{i}}\right)$ can be made arbitrarily big and so the expression is dominated by the lone $n$ and also diverges.