Problem 85 - Thomas Calculus in SI Units

Question

Show that $$\lim _{k \rightarrow \infty}\left(1+\frac{r}{k}\right)^{k}=e^{r}$$

Step-by-Step Solution

Answer

The limit evaluates to \(e^{r}\).

Step 1: Understand the Problem

We need to prove that \(\lim _{k \rightarrow \infty}\left(1+\frac{r}{k}\right)^{k}=e^{r}\). This limit is a representation of the exponential function \(e^{r}\) when viewed through the lens of continuous compounding in finance or repetitive growth in calculus.

Step 2: Use Known Limit Definition

Recognize the known limit \(\lim _{k \rightarrow \infty}\left(1+\frac{1}{k}\right)^{k}=e\). We can generalize this definition to show that \(\lim _{k \rightarrow \infty}\left(1+\frac{r/k}{k/r}\right)^{k/r}=e\), where the base is adjusted by a factor of \(r\).

Step 3: Substitute and Simplify

For \(\lim _{k \rightarrow \infty}\left(1+\frac{r}{k}\right)^{k}\), substitute \(\left(1+\frac{r}{k}\right)^{k}\) as \(\left(1+\frac{r}{k}\right)^{k/r \, \cdot\, r}\). Rewrite to utilize the property of exponents: \((a^{b})^{c} = a^{b \cdot c}\).

Step 4: Apply the Known Limit

Now, express the limit as \(\left(\lim _{k \rightarrow \infty} \left(1+\frac{r}{k}\right)^{k/r}\right)^{r}\). According to the known limit, inside the power becomes \(e\) as \(k \to \infty\). Thus the entire expression becomes \(e^{r}\).

Step 5: Conclusion

Conclude that \(\lim _{k \rightarrow \infty}\left(1+\frac{r}{k}\right)^{k}=e^{r}\). This follows by continuously repeating and compounding the fraction \(\frac{r}{k}\) as \(k\) grows large.

Key Concepts

Exponential Function

An exponential function in mathematics is crucial for modeling situations where growth or decay is rapid and continuous. It follows the formula \( f(x) = a \cdot b^x \), where \(a\) is a constant, \(b\) is the base, which is a positive real number, and \(x\) is the exponent. The most significant base in exponential functions is Euler's number \( e \approx 2.71828 \), often used in calculus due to its unique properties. The exponential function \( e^x \) is particularly important as it describes phenomena such as population growth, radioactive decay, and continuously compounded interest. In our exercise, \( e^r \) emerges naturally from the expression \( \lim_{k \to \infty} \left(1 + \frac{r}{k} \right)^k \), linking exponential growth with the concept of limits in calculus. By understanding exponential functions, students can appreciate how small, repeated additions or multiplications have large cumulative effects, which is a recurring theme in both mathematics and real-world scenarios.

Continuous Compounding

Continuous compounding is a financial concept that illustrates how an investment grows when interest is added back to the principal at every possible moment. While regular compounding might occur monthly, quarterly, or annually, continuous compounding occurs infinitely and is modeled by the exponential expression \( e^{rt} \).

In continuous compounding, as the frequency of compounding increases, it approaches an extreme where it is compounded at every instant. This limit gives rise to the exponential function, and specifically to the expression \( \lim_{k \to \infty} \left( 1 + \frac{r}{k} \right)^k \).

Understanding this concept is crucial for grasping how small changes can lead to significant changes over time. It emphasizes the significant impact of the constant \( e \) in real-world financial calculations.

Properties of Exponents

The properties of exponents are fundamental rules that simplify expressions involving powers, helping us manipulate and solve equations. These rules include:

• \(a^m \cdot a^n = a^{m+n}\)
• \((a^m)^n = a^{m \cdot n}\)
• \((ab)^n = a^n \cdot b^n\)
• \(a^{-n} = \frac{1}{a^n}\)

These properties are instrumental in solving the exercise given. When we simplify \( \left(1 + \frac{r}{k} \right)^{k} \) to its limit by rewriting it using the properties, specifically \((a^b)^c = a^{b \cdot c}\), we align it with known limits such as \( \lim_{k \to \infty} \left(1 + \frac{1}{k} \right)^k = e \).

By applying these properties, we can transform and understand complex limits more intuitively, showcasing the power of exponents in calculus and beyond.