Problem 58 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove that \(\lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{-n}=e\). Hint: First show that $$ \left(1-\frac{1}{n}\right)^{-n}=\left(1+\frac{1}{n-1}\right)^{n}=\left(1+\frac{1}{n-1}\right)^{n-1}\left(1+\frac{1}{n-1}\right) $$

Step-by-Step Solution

Answer

The limit is \(e\).

Step 1: Rewriting the Expression

We start by rewriting the expression \( \left(1-\frac{1}{n}\right)^{-n}\). Using the hint, we can express it as:\[ \left(1-\frac{1}{n}\right)^{-n} = \left(1+\frac{1}{n-1}\right)^{n} \]This transformation allows us to relate it to the exponential function.

Step 2: Apply Binomial Expansion

Next, consider the expression \( \left(1+\frac{1}{n-1}\right)^{n-1} \). For large \(n\), use the binomial theorem approximation:\[ \left(1+\frac{1}{n-1}\right)^{n-1} \approx e \]This approximation holds true as \(n\) approaches infinity since it resembles the definition of \(e\) as a limit.

Step 3: Evaluating the Remaining Factor

Now, for the remaining factor in the expression, consider:\[ \left(1+\frac{1}{n-1}\right) \rightarrow 1 \] as \(n \rightarrow \infty\). This factor approaches 1 since the increment becomes negligible as \(n\) grows.

Step 4: Combining Results

Combine both results from Steps 2 and 3:\[ \left(1+\frac{1}{n-1}\right)^{n} = \left(1+\frac{1}{n-1}\right)^{n-1} \cdot \left(1+\frac{1}{n-1}\right) \approx e \cdot 1 = e \]Thus, we prove that the original limit simplifies to \(e\) as \(n\) approaches infinity.

Key Concepts

Understanding the Exponential Function

The exponential function, usually represented as \(e^x\), is a fundamental concept in mathematics. Here, \(e\) is Euler's number, which is approximately 2.71828. This function describes growth that increases rapidly due to compounding, which is essential in many areas such as population growth, interest calculations, and natural phenomena.

A crucial property of the exponential function is that its rate of growth at any point is proportional to its current value. This forms the basis of the equation \(\frac{d}{dx}e^x = e^x\), which signifies that the derivative of the exponential function is itself. In the context of limits and sequences, the exponential function is related to the expression \(\left(1 + \frac{1}{n}\right)^n\), which approaches \(e\) as \(n\) becomes infinitely large. This is an example of how exponential behavior arises even in sequences aren't inherently exponential.

The Power of the Binomial Theorem

The binomial theorem provides a tool for expanding expressions that are raised to a power, such as \((x + y)^n\). It states:

• \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\)

Here, \(\binom{n}{k}\) represents the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\). This expansion shows that a binomial expression can be broken down into a sum of terms involving the binomial coefficients.

In applying the binomial theorem to limits, such as in our exercise, it helps approximate more complex expressions as sequences approach infinity. Using the theorem, one can simplify expressions that involve small fractions raised to powers, recognizing patterns and fundamental constants like \(e\). For very large values of \(n\), the binomial expansion of expressions like \(\left(1 + \frac{1}{n-1}\right)^{n-1}\) approaches \(e\), revealing its deep connection to the exponential function.

Exploring Infinite Limits

Infinite limits describe the behavior of functions or sequences as they approach infinity. The idea is to understand what value, if any, a sequence converges toward as \(n\) grows indefinitely large. For instance, in the given problem, the expression \(\left(1 - \frac{1}{n}\right)^{-n}\) transforms into \(\left(1 + \frac{1}{n-1}\right)^n\) to ultimately show it converges to \(e\).

When assessing infinite limits, it's often helpful to transform challenging expressions into forms that are recognizably navigable. Through comparisons, algebraic manipulation, or approximation techniques, one seeks to discover the sequence's ultimate behavior. Infinite limits are essential in calculus, helping to define and understand continuous growth and behavior in a variety of mathematical and real-world applications.

Crafting a Mathematical Proof

A mathematical proof is a logical argument that verifies the truth of a statement beyond any doubt. In mathematics, proofs are crucial, as they establish fundamental truths and help unravel complex problems into understandable solutions. Providing a proof involves:

• Clearly stating the problem or theorem.
• Breaking down the problem into smaller, manageable parts.
• Applying relevant mathematical principles or theorems.
• Building a logical sequence of statements leading to the conclusion.

In the exercise of proving the limit \(\left(1 - \frac{1}{n}\right)^{-n} = e\), each step of the solution systematically addresses changes through substitutions and approximations, ultimately converging to the exponential constant \(e\). The work reinforces understanding not just of the result but also of why and how the expression behaves as it does. Approaching problems this way ensures that every claim in mathematics is underpinned by rigorous, verifiable logic.