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Use Bernoulli's Inequality to show that lim as n tends to infinity for 2^1/n = 1.

User Blexy
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Final Answer:

Using Bernoulli's Inequality, we can show that as n tends to infinity, the limit of 2^(1/n) is 1.

Step-by-step explanation:

Bernoulli's Inequality states that for any real number x > -1 and any natural number n, (1 + x)^n ≥ 1 + nx. Applying this inequality to our expression, where x = -1/2 and n is a positive integer, we get (1 - 1/2)^n = 1/2^n. According to Bernoulli's Inequality, 1/2^n is greater than or equal to 1 - n/2. Rearranging, we have 1/2^n ≥ 1 - n/2.

As n tends to infinity, the term n/2 becomes negligible compared to 1, and the right side of the inequality approaches 1. Therefore, 1/2^n approaches 1 as n tends to infinity. This implies that the limit of 2^(1/n) as n approaches infinity is indeed 1.

In conclusion, by employing Bernoulli's Inequality and analyzing the behavior of the given expression, we have established that the limit of 2^(1/n) tends to 1 as n tends to infinity. This application of Bernoulli's Inequality provides a concise and rigorous way to demonstrate the convergence of the sequence to the desired limit.

User Alex Keil
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