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Question Number 197947 by mnjuly1970 last updated on 05/Oct/23

$$$$$$\mathrm{calculate}\dots$$$$\mathrm{L}=\lim {}_{\mathrm{n}\to \mathrm{\infty }}\sqrt[n]{(1+\frac{1}{2})(1+\frac{1}{3})\dots (1+\frac{1}{n})}=?$$$$$$

Answered by MM42 last updated on 05/Oct/23

$$A={lim}_{n\to \mathrm{\infty }}{(\frac{3}{2}\times \frac{4}{3}\times ...\times \frac{n}{n-1}\times \frac{n+1}{n})}^{\frac{1}{n}}$$$$lnA={lim}_{n\to \mathrm{\infty }}ln{\left(\frac{n+1}{2}\right)}^{\frac{1}{n}}$$$$$$$$={lim}_{n\to \mathrm{\infty }}\frac{ln\left(\frac{n+1}{2}\right)}{n}=0\Rightarrow A=1✓$$$$$$

Commented by mnjuly1970 last updated on 05/Oct/23

$$thxalotsir...$$

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