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

calculate \dots

L = \lim_{n \to \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 = {l i m}_{n \to \infty} {(\frac{3}{2} \times \frac{4}{3} \times \dots \times \frac{n}{n - 1} \times \frac{n + 1}{n})}^{\frac{1}{n}}

l n A = {l i m}_{n \to \infty} l n {(\frac{n + 1}{2})}^{\frac{1}{n}}

= {l i m}_{n \to \infty} \frac{l n (\frac{n + 1}{2})}{n} = 0 \Rightarrow A = 1 ✓

Commented by mnjuly1970 last updated on 05/Oct/23

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