(2n) /n^(2n)

Jasonbradley

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Jun 16, 2024

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Exploring the Limit of (2n) / n^(2n)

This article delves into the behavior of the sequence (2n) / n^(2n) as n approaches infinity. We'll explore how this sequence behaves and determine its limit.

Understanding the Expression

The expression (2n) / n^(2n) represents a ratio where the numerator grows linearly with n, while the denominator grows exponentially. This exponential growth in the denominator suggests that the ratio might become increasingly small as n gets larger.

Analyzing the Limit

To formally determine the limit, we can use the following steps:

1. Rewrite the expression: (2n) / n^(2n) can be rewritten as 2 / (n^(2n-1)). This emphasizes the exponential growth of the denominator.

2. Apply limit laws: As n approaches infinity, the term (n^(2n-1)) becomes infinitely large. Therefore, the entire expression approaches 0.

Formal Proof

We can use the squeeze theorem to prove this formally:

1. Bound the expression: Since n > 1, we have 0 < (2n) / n^(2n) < 2 / n.

2. Apply limits: As n approaches infinity, both 0 and 2/n approach 0.

3. Conclude using squeeze theorem: Since the expression is bounded between two sequences that converge to 0, the expression (2n) / n^(2n) also converges to 0 as n approaches infinity.

Conclusion

The sequence (2n) / n^(2n) approaches 0 as n approaches infinity. This demonstrates the dominance of exponential growth over linear growth, highlighting the power of exponential functions in the context of limits.