Question

Use Taylor's theorem to derive the binomial theorem: $$(1+x)^{n}=\left(\begin{array}{l}n \\\0\end{array}\right)+\left(\begin{array}{l}n \\\1\end{array}\right) x+\left(\begin{array}{l}n \\\2\end{array}\right) x^{2}+\cdots+\left(\begin{array}{l}n \\\n\end{array}\right) x^{n}$$\ Here the binomial coefficients \(\left(\begin{array}{c}n \\\ r\end{array}\right)\) are defined by $$\left(\begin{array}{l}n \\\r\end{array}\right)=\frac{n !}{r !(n-r) !}$$ where \(n !=n(n-1) \cdots 2 \cdot 1\) if \(n \geq 1\) and \(0 !=1\).

Short answer

The binomial theorem is derived by expanding \((1+x)^n\) using Taylor's series and recognizing the coefficients as binomial coefficients.

Step-by-step solution

Understand The Function

We need to find the Taylor series expansion of \( f(x) = (1+x)^{n} \) around \( x = 0 \). This requires finding the derivatives of \( f(x) \) at \( x = 0 \). The Taylor series is given by: \[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(k)}(0)}{k!}x^k + \cdots \]

Calculate the Function Value at x = 0

Evaluate \( f(x) = (1+x)^{n} \) at \( x = 0 \):\[ f(0) = (1+0)^n = 1 \]

Find the First Derivative

The first derivative of \( f(x) = (1+x)^{n} \) is:\[ f'(x) = n(1+x)^{n-1} \] Evaluate at \( x = 0 \):\[ f'(0) = n(1+0)^{n-1} = n \]

Find the Second Derivative

The second derivative is obtained by differentiating again:\[ f''(x) = n(n-1)(1+x)^{n-2} \] Evaluate at \( x = 0 \):\[ f''(0) = n(n-1)(1+0)^{n-2} = n(n-1) \]

Generalize the k-th Derivative

The pattern for the k-th derivative is:\[ f^{(k)}(x) = n(n-1)(n-2)...(n-k+1)(1+x)^{n-k} \]Evaluate this derivative at \( x = 0 \):\[ f^{(k)}(0) = n(n-1)(n-2)...(n-k+1) \]

Write the Taylor Series

Substitute these results into the Taylor series expression:\[ (1+x)^{n} = 1 + nx + \frac{n(n-1)}{2!}x^2 + \cdots + \frac{n(n-1)...(n-k+1)}{k!}x^k + \cdots \]

Introduction of Binomial Coefficients

Recognize that the coefficients\[ \frac{n(n-1)...(n-k+1)}{k!} \] can be written using binomial coefficients as\[ \left(\begin{array}{c}n \ k\end{array}\right) = \frac{n!}{k!(n-k)!} \]Thus, the expansion becomes:\[ (1+x)^{n} = \sum_{k=0}^{n} \left(\begin{array}{c}n \ k\end{array}\right)x^k \]

Conclusion of the Proof

Therefore, using Taylor's theorem and recognizing the defined binomial coefficients, we successfully derive the binomial expansion:\[ (1+x)^{n} = \sum_{k=0}^{n} \left(\begin{array}{c}n \ k\end{array}\right)x^k \]

Key concepts

Taylor Series

A 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. In this exercise, we focus on expanding the function \( f(x) = (1+x)^n \) around the point \( x = 0 \). The Taylor series formula is given by:\[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(k)}(0)}{k!}x^k + \cdots \]This method helps in approximating functions using polynomials and is especially useful when a function cannot be expressed easily in regular algebraic terms. By focusing on the function and its derivatives evaluated at zero, we can construct the series to predict how the function behaves locally.

Binomial Coefficients

The binomial coefficients are a central part of the binomial theorem, which helps express powers of sums. The coefficient denoted by \( \left(\begin{array}{c}n \ r\end{array}\right) \) is calculated as:\[\left(\begin{array}{l}n \ r\end{array}\right) = \frac{n!}{r!(n-r)!}\]These coefficients appear in the expression of a binomial expansion. When we expand \((1+x)^n\), each term is multiplied by a binomial coefficient denoting the number of ways to choose \( r \) items from \( n \).

• They arise naturally when considering the expansion due to their combinatorial significance.
• They represent the weights of the terms in the polynomial expansion.

Using binomial coefficients, each component of the expanded series can be easily calculated, making them a powerful tool in algebra.

Derivatives

A derivative measures how a function changes as its input changes. Calculating derivatives is crucial for constructing the Taylor series of a function. For our function \( f(x) = (1+x)^n \), we need to determine its derivatives at \( x = 0 \). This includes:- **First derivative**: \( f'(x) = n(1+x)^{n-1} \). Evaluated at zero gives \( f'(0) = n \).- **Second derivative**: \( f''(x) = n(n-1)(1+x)^{n-2} \). At zero, it evaluates to \( f''(0) = n(n-1) \).This pattern continues for higher derivatives. Each step involves reducing the power and multiplying by additional factors derived from the power of \( x \) in the original function.The derivatives help form the Taylor series by determining the coefficients that define how rapidly the function changes, leading directly to the polynomial approximations of the function.

Mathematical Proof

Mathematical proofs are structured logical arguments that demonstrate the truth of mathematical statements. Proofs use definitions, axioms, and previously established results to arrive at a conclusion. In proving the binomial theorem using Taylor's theorem, we:

• Start by expressing \( f(x) = (1+x)^n \) using the Taylor series.
• For each derivative evaluated at zero, recognize the resulting series elements.
• Introduce binomial coefficients to simplify these series terms into the familiar form.

The proof concludes by showing that the derived expression using the Taylor series matches the known binomial expansion. This rigorous approach confirms the relation and helps understand not just the result, but the process to reach the statement \( (1+x)^{n} = \sum_{k=0}^{n} \left(\begin{array}{c}n \ k\end{array}\right)x^k \). Such proofs are essential for validating mathematical theories and their applications.