Problem 5 - Basic Analysis: Introduction to Real Analysis

Question

Show $$ e^{x}=\lim _{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^{n} $$ Hint: Take the logarithm. Note: The expression \(\left(1+\frac{x}{n}\right)^{n}\) arises in compound interest calculations. It is the amount of money in a bank account after 1 year if 1 dollar was deposited initially at interest \(x\) and the interest was compounded n times during the year. The exponential \(e^{x}\) is the result of continuous compounding.

Step-by-Step Solution

Answer

Taking the logarithm and using series expansion, \( \lim_{n \rightarrow \infty} \left(1+\frac{x}{n}\right)^{n} = e^{x} \).

Step 1: Define the Sequence

We start with the expression \( a_n = \left(1 + \frac{x}{n}\right)^n \) and our goal is to evaluate \( \lim_{n \to \infty} a_n \) and show that it equals \( e^x \).

Step 2: Apply the Logarithm

Taking the natural logarithm of both sides gives \( \ln(a_n) = n \ln\left(1 + \frac{x}{n}\right) \). This step helps in simplifying the limit process by using properties of logarithms.

Step 3: Expand Using Taylor Series

Use the Taylor series expansion for \( \ln(1 + u) \approx u - \frac{u^2}{2} + \frac{u^3}{3} - \cdots \), where \( u = \frac{x}{n} \). The leading term for large \( n \) is \( \frac{x}{n} \).

Step 4: Simplify the Logarithmic Expression

Substitute the first term of the expansion into the logarithm: \( n \ln\left(1 + \frac{x}{n}\right) \approx n\left(\frac{x}{n}\right) = x \). Other terms tend to zero as \( n \to \infty \).

Step 5: Exponentiate to Find Limit

Since we have \( \ln(a_n) \to x \), exponentiating both sides gives \( a_n = \left(1 + \frac{x}{n}\right)^n \to e^x \) as \( n \to \infty \). Thus, we show that \( e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n \).

Key Concepts

Natural Logarithm

The Natural Logarithm, often denoted as \ln(x)\, is a logarithm that has a special base, the number \e\. The number \e\ is approximately 2.71828 and is an irrational constant often used in mathematical analyses, particularly in calculus. The natural log is used primarily for simplifying exponential expressions, which is essential when dealing with limits and compound interest calculations.
The natural logarithm has properties that make it particularly handy:

• \(\ln(ab) = \ln(a) + \ln(b)\)
• \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
• \(\ln(a^b) = b \ln(a)\)

Applying the natural logarithm makes certain algebraic manipulations easier, as seen in this limit proof of the exponential function. Here, taking the natural log of both sides allows us to tackle the exponent and break down the problem using logarithmic properties.

Taylor Series

The Taylor Series is a mathematical tool used to approximate functions. It represents functions as an infinite sum of terms calculated from the values of its derivatives at a single point. The Taylor Series expansion for natural logarithms gives us an approximate equation to use:
\[ \ln(1 + u) \approx u - \frac{u^2}{2} + \frac{u^3}{3} - \cdots \]
By using the Taylor Series, we can simplify complex functions such as \(\ln\left(1 + \frac{x}{n}\right)\).
This method is particularly useful when tackling the exponential limit proof since higher order terms tend to zero as \ increases, allowing us to focus only on significant terms that affect the limit.
Using just the leading terms makes calculations simpler and shows convergence as seen in continuous compounding studies.

Compound Interest

Compound Interest is a core concept in financial mathematics. It refers to the way interest is calculated on both the initial principal and the accumulated interest from previous periods. When interest is compounded multiple times per year, the amount grows faster than with simple interest.
A formula often used for compound interest is:

• \[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]

Where:

• \(A\) = the amount of money accumulated after n years, including interest.
• \(P\) = principal amount (the initial amount).
• \(r\) = annual interest rate (as a decimal).
• \(n\) = number of times interest is compounded per year.
• \(t\) = time the money is invested for in years.

In the context of an exponential limit like \(\left(1 + \frac{x}{n}\right)^n\), it represents the theoretical situation of what happens as interest is compounded more frequently.

Continuous Compounding

Continuous Compounding reflects a hypothetical situation where interest is compounded an infinite number of times per year, leading to exponential growth.
The formula used for calculating the future value under continuous compounding is:

• \(A = Pe^{rt}\)

This gives a more precise value than regular compounding calculations as it considers the limit as \(n\) approaches infinity.
The exponential \(e^x\) is the most familiar result of continuous compounding. In practical terms, it's used to figure out the ultimate increase in capital when considering continuous growth factors. In the exponential limit proof, we explore how consistent compounding leads to an increasing function, \(e^x\), which showcases the amazing limit behavior of exponential functions.