%5En-converge-or-diverge.jpeg "(1+2/n)^n Converge Or Diverge")
The Limit of (1 + 2/n)^n: Convergence or Divergence?
The expression (1 + 2/n)^n is a fascinating one in mathematics, as it explores the concept of convergence and divergence in sequences. Let's delve into the analysis to determine whether this sequence converges to a finite value or diverges to infinity as 'n' approaches infinity.
Understanding Convergence and Divergence
- Convergence: A sequence converges if its terms get arbitrarily close to a specific value as 'n' increases indefinitely. This value is called the limit of the sequence.
- Divergence: A sequence diverges if its terms do not approach a finite value as 'n' increases. This can happen in different ways, such as tending towards infinity or oscillating without settling on a specific value.
Analyzing the Limit of (1 + 2/n)^n
We can employ several methods to analyze the limit of (1 + 2/n)^n as 'n' approaches infinity:
1. Intuition and Observation:
As 'n' gets larger, the term 2/n becomes smaller. Therefore, (1 + 2/n) approaches 1. However, we are raising this value to a power that is also growing infinitely large. This interplay between a value approaching 1 and an increasingly large exponent suggests that the limit might exist but is not immediately obvious.
2. Using a Limit Theorem:
A well-known theorem in calculus states:
lim (1 + x/n)^n = e^x as n approaches infinity.
This theorem is directly applicable to our problem if we substitute x = 2. Therefore:
lim (1 + 2/n)^n = e^2 as n approaches infinity.
3. Numerical Exploration:
We can use a calculator or a computer program to calculate the values of (1 + 2/n)^n for increasingly large values of 'n'. This will show that the values get closer and closer to a specific value, approximately 7.389.
Conclusion
Based on the analysis above, we can conclude that the sequence (1 + 2/n)^n converges to e^2, which is approximately 7.389. This signifies that as 'n' approaches infinity, the terms of the sequence get arbitrarily close to this value, demonstrating the behavior of convergence.