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This question already has an answer here:

Let $$a_n=\left(1-\dfrac{1}{\sqrt2}\right)\dots \left(1-\dfrac{1}{\sqrt{n+1}}\right),n\ge1$$ Then find $\lim_{n\to \infty} a_n$.

How can I proceed? I am stuck at the first step. Please help.

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marked as duplicate by quid Feb 25 at 14:36

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Note that $a_n>0$ and $$\lim_{n\to\infty}\log a_n=\sum_{k=1}^\infty \log \left(1-\dfrac{1}{\sqrt{k+1}}\right)$$ is divergent to $-\infty$ by comparaison test with Riemann divergent series: $$\log \left(1-\dfrac{1}{\sqrt{k+1}}\right)\sim_\infty -\dfrac{1}{\sqrt{k}}$$ hence $$\lim_{n\to\infty} a_n=0$$

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If you consider $b_n=\ln a_n$, what can you say about the behavior of the sequence $(b_n)_{n\geq 2}$?

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