%5En-limit.jpeg "(1+2/n)^n Limit")
The Limit of (1 + 2/n)^n as n Approaches Infinity
In calculus, we often encounter limits that involve sequences or functions. One particularly interesting and important limit is that of (1 + 2/n)^n as n approaches infinity. This limit has a significant connection to the exponential function and is used in various applications like compound interest and probability theory.
Understanding the Limit
As 'n' gets larger and larger, the term '2/n' becomes smaller and smaller. However, the exponent 'n' is also increasing. This interplay between the shrinking base and increasing exponent is what makes the limit intriguing.
Finding the Limit
The limit of (1 + 2/n)^n as n approaches infinity is e^2, where 'e' is the mathematical constant approximately equal to 2.71828.
Here's how we can approach finding this limit:
- Using the definition of 'e': The constant 'e' is defined as the limit of (1 + 1/n)^n as n approaches infinity. We can use this definition to rewrite our limit:
lim (n→∞) (1 + 2/n)^n = lim (n→∞) [(1 + 1/(n/2))^n]
Now, let's substitute 'm = n/2'. As 'n' approaches infinity, 'm' also approaches infinity.
lim (m→∞) [(1 + 1/m)^(2m)]
We can now use the power of a power property to separate the exponent:
lim (m→∞) [(1 + 1/m)^m]^2
From the definition of 'e', we know that (1 + 1/m)^m approaches 'e' as 'm' approaches infinity. Therefore:
lim (m→∞) [(1 + 1/m)^m]^2 = e^2
- Using the Binomial Theorem: We can expand the expression (1 + 2/n)^n using the Binomial Theorem:
(1 + 2/n)^n = 1 + n*(2/n) + (n*(n-1)/2!)*(2/n)^2 + ... + (2/n)^n
Simplifying and taking the limit as n approaches infinity, we can observe that the terms after the first few become negligible. The first few terms converge to the series expansion of e^2.
Importance of the Limit
This limit is crucial in various mathematical concepts and applications:
- Compound Interest: The limit represents the continuous compounding of interest where the interest is compounded infinitely many times per year.
- Probability Theory: The limit plays a role in the Poisson distribution, which models the probability of a certain number of events occurring in a fixed interval of time or space.
Conclusion
The limit of (1 + 2/n)^n as n approaches infinity is a significant mathematical concept with connections to the exponential function and various applications in different fields. Understanding the limit and its derivation provides insights into the behavior of exponential growth and allows for more precise calculations in various applications.