Calcualte the limit of $\frac{\sqrt n!}{(1+\sqrt 1)(1+\sqrt 2)...(1+\sqrt n)}$ as n approaches infinity

Asked Dec 14 '22 at 22:01

Active Dec 14 '22 at 22:01

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$ \lim_{n \to +\infty}\frac{\sqrt n!}{(1+\sqrt 1)(1+\sqrt 2)*...*(1+\sqrt n)}$
Any hints / solutions here? I feel like I'm stuck. The Ratio test is equal to $1$ which doesn't help at all. Intuitively, I'd say the limit of this series is $0$ but I can't think of how I would squeeze the series if the limit is in fact $0$.

Preferably no L'Hopital, as we haven't learned that yet (I don't really know if L'Hopital is even applicable in this case).

asked Dec 14 '22 at 22:01

cezar

3

Is it $\sqrt n!$ or $\sqrt{n!}$ in the numerator? – Martin R Dec 14 '22 at 22:12
2

take logarithms and estimate the pieces. – Will Jagy Dec 14 '22 at 22:12

Calcualte the limit of $\frac{\sqrt n!}{(1+\sqrt 1)(1+\sqrt 2)*...*(1+\sqrt n)}$ as n approaches infinity

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Calcualte the limit of $\frac{\sqrt n!}{(1+\sqrt 1)(1+\sqrt 2)...(1+\sqrt n)}$ as n approaches infinity