# Find the value of $\lim_{n \rightarrow \infty} \Big( 1-\frac{1}{\sqrt 2} \Big) \cdots \Big(1-\frac{1}{\sqrt {n+1}} \Big)$ [duplicate]

$$\lim_{n \rightarrow \infty} \Big( 1-\dfrac{1}{\sqrt 2} \Big) \cdots \Big(1-\dfrac{1}{\sqrt {n+1}} \Big)$$

Attempt:

Let $y = \lim_{n \rightarrow \infty} \Big( 1-\dfrac{1}{\sqrt 2} \Big) \cdots \Big(1-\dfrac{1}{\sqrt {n+1}} \Big)$

$\ln y = \lim_{n \rightarrow \infty} \ln \Big( 1-\dfrac{1}{\sqrt 2} \Big)+ \cdots + \ln \Big(1-\dfrac{1}{\sqrt {n+1}} \Big)$

I am not able to move ahead really from here. Could someone give me an hint on how to move forward with this problem.

Thank you very much for your help in this regard.

## marked as duplicate by Sil, mrtaurho, Delta-u, ancientmathematician, max_zornFeb 24 at 21:55

All the factors are positive, but bounded by the last one. Thus, for all $n$ the product is between $0$ and $(1-1/\sqrt{n+1})^n$. What happens with that expression as $n\to+\infty$?
• The limit tends to $0$. – MathMan Apr 30 '15 at 11:46
• The limit is $0$, or the sequence tends to $0$. – GEdgar Apr 30 '15 at 12:28