%5En-limit.jpeg "(1-1/n)^n Limit")
Understanding the Limit of (1 - 1/n)^n
In calculus, the limit of a sequence is a fundamental concept. One particularly intriguing example is the limit of the sequence (1 - 1/n)^n as n approaches infinity. This sequence has a fascinating connection to a fundamental mathematical constant: e.
Defining the Limit
The limit of a sequence (a_n) as n approaches infinity is the value that the sequence approaches as n becomes infinitely large. We denote this limit as:
lim (n→∞) a_n
In our case, we want to find:
lim (n→∞) (1 - 1/n)^n
Exploring the Limit
Let's explore the behavior of the sequence as n increases:
- n = 1: (1 - 1/1)^1 = 0
- n = 2: (1 - 1/2)^2 = 0.25
- n = 10: (1 - 1/10)^10 ≈ 0.3487
- n = 100: (1 - 1/100)^100 ≈ 0.3660
- n = 1000: (1 - 1/1000)^1000 ≈ 0.3679
As n gets larger, the sequence appears to be approaching a value close to 0.3679.
The Connection to e
Surprisingly, the limit of (1 - 1/n)^n as n approaches infinity is not 0.3679. Instead, it converges to a very important mathematical constant known as e, which is approximately 2.71828.
This might seem counterintuitive, but the sequence approaches e from below, getting closer and closer as n gets larger.
Proving the Limit
Proving that the limit of (1 - 1/n)^n is e requires a more advanced understanding of calculus and uses techniques like L'Hopital's rule and Taylor series. The proof involves showing that the limit of the sequence is equal to the limit of a different sequence that is known to converge to e.
Significance of the Limit
The limit of (1 - 1/n)^n is crucial in various fields, including:
- Calculus: It plays a role in the definition of the natural logarithm and exponential functions.
- Finance: The formula is used in calculating compound interest when the interest is compounded continuously.
- Probability and Statistics: The limit appears in probability distributions, such as the Poisson distribution.
Conclusion
The limit of (1 - 1/n)^n as n approaches infinity is a fascinating example of a sequence converging to a constant. Its connection to the mathematical constant e highlights the interconnectedness of different areas of mathematics and its impact across various fields. While understanding the proof requires a deeper mathematical background, the intuition behind the limit and its significance are valuable insights.