Problem 1 - A Modern Approach to Quantum Mechanics

Question

Show that $$ \lim _{N \rightarrow \infty}\left(1+\frac{x}{N}\right)^{N}=e^{x} $$ by comparing the Taylor series expansions for the two functions.

Step-by-Step Solution

Answer

The Taylor series of both functions match as \(N\) approaches infinity.

Step 1: Write the Taylor Series for \(e^{x}\)

The Taylor series expansion of \(e^{x}\) around \(x = 0\) is given by:\[e^{x} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\]

Step 2: Expand \(\left(1 + \frac{x}{N}\right)^N\) Using Binomial Theorem

Use the binomial theorem to expand:\[\left(1 + \frac{x}{N}\right)^N = \sum_{k=0}^{N} \binom{N}{k} \left(\frac{x}{N}\right)^k\]This gives us:\[= 1 + \binom{N}{1} \frac{x}{N} + \binom{N}{2} \left(\frac{x}{N}\right)^2 + \cdots\]

Step 3: Simplify the Binomial Coefficients

The terms simplify as follows:- For \(k=1\), \(\binom{N}{1} = N\) so the term becomes \(x\).- For \(k=2\), \(\binom{N}{2} = \frac{N(N-1)}{2}\), so the term becomes \(\frac{x^2}{2!}\) as \(N\) approaches infinity.- Continue this pattern for higher terms.

Step 4: Compare the Expansions

As \(N\) approaches infinity, the terms of \(\left(1 + \frac{x}{N}\right)^N\) converge to the corresponding terms in the Taylor expansion of \(e^{x}\):- First term: 1- Second term: \(x\)- Third term: \(\frac{x^2}{2!}\)- And so on...This shows that the expansion of \(\left(1 + \frac{x}{N}\right)^N\) closely matches the Taylor series of \(e^{x}\) as \(N\) becomes very large.

Key Concepts

Taylor Series Expansion

The Taylor Series is a way to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point. Think of it as breaking down a complex function into an infinite series of simpler polynomial terms. For many transcendent functions, such as the exponential function, the Taylor Series is particularly useful.In our exercise, the Taylor Series for the function \(e^{x}\) is given by the equation:

• \(e^{x} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\)

Here, the exclamation mark '!' represents a factorial, meaning you multiply the number by every positive integer less than it. For instance, \(3! = 3 \times 2 \times 1 = 6\). Each term in this series is derived based on the derivatives of \(e^{x}\) evaluated at 0. The beauty of this series is its convergence property over the entire real line for \(e^{x}\), making it an excellent tool for approximations in calculus.

Binomial Theorem

The Binomial Theorem provides a powerful method to expand expressions that are raised to a power. It's particularly handy when dealing with expressions like \(\left(1 + \frac{x}{N}\right)^N\). According to the Binomial Theorem, this can be expanded as follows:

• \(\left(1 + \frac{x}{N}\right)^N = \sum_{k=0}^{N} \binom{N}{k} \left(\frac{x}{N}\right)^k\)

Here, the term \(\binom{N}{k}\) represents a binomial coefficient, which is a specific way of selecting \(k\) items from \(N\), mathematically given as \(\frac{N!}{k!(N-k)!}\). This expansion is crucial for understanding how an expression behaves as \(N\) grows very large, breaking the expression into simpler sums that we can compare to other series like the Taylor series of \(e^{x}\). As \(N\) becomes large, the terms of the binomial expansion approximate more closely to those of the Taylor Series of \(e^{x}\). This forms the basis for many proofs involving continuous growth processes and compound interest calculations.

Convergence of Series

Convergence of a series refers to the approach of its sum to a certain value as the number of terms increases. In our context, it's the process by which the expanded expression \(\left(1 + \frac{x}{N}\right)^N\) nears the Taylor Series representation of \(e^{x}\). When we say a series converges, we mean that as more terms are added, the series approaches a specific value, or "converges" into a particular point.

• As \(N\) becomes very large, each segment in the binomial expansion \(\left(1 + \frac{x}{N}\right)^N\) looks more like its counterpart in the series for \(e^{x}\).

In our original problem, convergence is key to demonstrating that the limit of the binomial expression as \(N\) approaches infinity is indeed \(e^{x}\). This is a profound concept in calculus and analysis, underpinning many of its core principles, such as using series to approximate functions and evaluate continuous growth.