13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.


{√((1 + 1 / 2n)ⁿ)}

Verified step by step guidance

1. Identify the given sequence: \(a_n = \sqrt{\left(1 + \frac{1}{2n}\right)^n}\).
2. Rewrite the sequence to a form that is easier to analyze by expressing the square root as a power: \(a_n = \left(1 + \frac{1}{2n}\right)^{\frac{n}{2}}\).
3. Recognize that the expression inside the parentheses resembles the form \(\left(1 + \frac{1}{m}\right)^m\) which is related to the number \(e\) as \(m \to \infty\).
4. Set \(m = 2n\) so that the expression becomes \(\left(1 + \frac{1}{m}\right)^{\frac{m}{2}}\) and analyze the limit as \(m \to \infty\).
5. Use the known limit \(\lim_{m \to \infty} \left(1 + \frac{1}{m}\right)^m = e\) to conclude that \(\lim_{n \to \infty} a_n = e^{\frac{1}{2}}\).

Key Concepts

Limits of Sequences

The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.

Exponential and Root Functions in Limits

When sequences involve expressions like powers and roots, it is important to simplify or rewrite them using properties of exponents and radicals. This often helps in identifying the behavior of the sequence as the index grows large.

Limit of (1 + 1/n)^n as n Approaches Infinity

The sequence (1 + 1/n)^n is a classic limit that approaches the mathematical constant e (~2.718). Recognizing this limit helps in evaluating more complex sequences that include similar expressions raised to powers.