Product of infinite sums

By the formula of sum of infinite geometric series, we have

$S_a = 1 + \frac{1}{a} + \frac{1}{a^2} + \frac{1}{a^3} + \cdots \Rightarrow S_a = \frac{1}{1-\frac{1}{a}} \Rightarrow S_a = \frac{a}{a-1}$ .

Checking $\lambda$ , we see that the product telescopes:

$\lambda = 2 \times \frac{3}{2} \times \cdots \times \frac{2012}{2011} \times \frac{2013}{2012} \Rightarrow \lambda = 2013$ .

Therefore, $\ \boxed{? = 1.}$

We know that $S_a=\frac{a}{a-1}$ . That means that $\prod_{2}^{2013}S_i=\frac{2}{1}\frac{3}{2}\frac{4}{3}...\frac{2012}{2011}\frac{2013}{2012}= 2013=\lambda$ .So $\frac{\lambda}{2013}=\boxed1$