Find the value of $\lim\limits_{n \to \infty} \left(1+\frac{C_0}{C_1}\right)\left(1+\frac{C_1}{C_2}\right)\cdots\left(1+\frac{C_{n-1}}{C_n}\right).$

Question

Let $(1 + x)^n = C_0 + C_1 x + \cdots + C_n x^n.$ Then find the value of $$\lim\limits_{n \to \infty} \left(1+\frac{C_0}{C_1}\right)\left(1+\frac{C_1}{C_2}\right)\cdots\left(1+\frac{C_{n-1}}{C_n}\right).$$

I find that $$\left(1+\frac{C_0}{C_1}\right)\left(1+\frac{C_1}{C_2}\right)\cdots\left(1+\frac{C_{n-1}}{C_n}\right) = \frac {(n+1)^{n}} {n!}.$$

Now since $n! \lt n^{n-1}$ for all $n \gt 2$ it follows that the sequence is divergent. So the required value should be infinity. But can I write the value to be infinity as infinity is not a real number? Any suggestion in this regard would be warmly appreciated.

Thanks a bunch.