let $a_n=(1-{1\over \sqrt2 })(1-{1\over \sqrt3 })\cdots (1-{1\over \sqrt {n+1} }), n\ge1$ then find $\lim_{n\to \infty}a_n$

Asked Nov 15 '22 at 14:34

Active Nov 15 '22 at 17:15

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My attempt : $\log a_n=\log(1-{1\over \sqrt2 })+\log(1-{1\over \sqrt3 })+\cdots +\log(1-{1\over \sqrt {n+1} })$ then how can I bring this into $\lim _{n\to \infty}{1\over n} \sum_{r=1}^n f({r\over n })$. please help. if there are any other way to approach do mention. Thanks in Advance.

asked Nov 15 '22 at 14:34

Chris

1

Use the asymptotics $$\log(1+t) \sim t$$ and conclude that $\log a_n \to - \infty$. This means that $a_n \to 0$. – Crostul Nov 15 '22 at 14:38
See also https://math.stackexchange.com/q/3124615/42969 and the linked questions. – Martin R Nov 15 '22 at 14:53

let $a_n=(1-{1\over \sqrt2 })(1-{1\over \sqrt3 })\cdots (1-{1\over \sqrt {n+1} }), n\ge1$ then find $\lim_{n\to \infty}a_n$

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