Problem 32 - Calculus for Biology and Medicine

Question

We can show that the Taylor polynomial for \(f(x)=(1+x)^{\alpha}\) about \(x=0\), with \(\alpha\) a positive constant, converges for \(x \in(-1,1)\). Show that $$ \begin{aligned} (1+x)^{\alpha}=& 1+\alpha x+\frac{\alpha(\alpha-1)}{2 !} x^{2} \\ &+\frac{\alpha(\alpha-1)(\alpha-2)}{3 !} x^{3}+\cdots+R_{n+1}(x) \end{aligned} $$

Step-by-Step Solution

Answer

The Taylor series \( (1+x)^{\alpha} = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3 + \cdots \) includes a remainder \( R_{n+1}(x) \) and converges for \( x \in (-1,1) \).

Step 1: Identify the Function and its Derivatives

The function given is \( f(x) = (1+x)^{\alpha} \). The Taylor series is based on derivatives of the function. Let's calculate the first few derivatives:\1. \( f(x) = (1+x)^{\alpha} \)\2. First derivative: \( f'(x) = \alpha (1+x)^{\alpha - 1} \)\3. Second derivative: \( f''(x) = \alpha(\alpha-1)(1+x)^{\alpha - 2} \)\4. Third derivative: \( f^{(3)}(x) = \alpha(\alpha-1)(\alpha-2)(1+x)^{\alpha - 3} \)\Continue this pattern to find higher derivatives.

Step 2: Set Up the Taylor Series

The general formula for the Taylor series of \( f(x) \) around \( x = 0 \) is given by: \[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f^{(3)}(0)}{3!}x^3 + \cdots \] Substitute the derivatives evaluated at \( x = 0 \). Calculating these, we get:- \( f(0) = 1^{\alpha} = 1 \)- \( f'(0) = \alpha \times 1^{\alpha - 1} = \alpha \)- \( f''(0) = \alpha(\alpha-1) \times 1^{\alpha - 2} = \alpha(\alpha-1) \)- \( f^{(3)}(0) = \alpha(\alpha-1)(\alpha-2) \times 1^{\alpha - 3} = \alpha(\alpha-1)(\alpha-2) \)

Step 3: Write Down the Taylor Polynomial

Insert the evaluated derivatives into the Taylor series:\[ f(x) = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3 + \cdots \] This matches the provided expansion, confirming the structure of the Taylor series.

Step 4: Consider the Remainder Term

The remainder term in the Taylor series, denoted as \( R_{n+1}(x) \), accounts for the error in approximating \( f(x) \) by the finite Taylor polynomial. For a series with a radius of convergence \((x \in (-1,1))\), this term approaches zero as \( n \) increases, ensuring the series converges to \( f(x) \). This confirms that our polynomial represents \((1+x)^{\alpha}\) adequately within the interval.

Key Concepts

Polynomial approximation

Polynomial approximation is a crucial concept in understanding Taylor series. It involves approximating a more complex function using a polynomial whose coefficients are derived from the function's derivatives evaluated at a specific point. For instance, the function \( f(x) = (1+x)^{\alpha} \) can be approximated using a polynomial around \( x = 0 \). Just like in the exercise, such a representation allows us to simplify calculations and make predictions about the behavior of \( f(x) \) without evaluating the entire function.
This polynomial is constructed by evaluating the function's derivatives and gradually adding terms to improve the approximation. The more terms added, the more accurate the approximation will be over its interval of convergence. Polynomial approximations help in solving complex problems by reducing them to manageable expressions. They are also essential for numerical computations in engineering, physics, and computer science.

Derivative evaluation

Derivative evaluation is an essential step when constructing a Taylor series. In the exercise, the first few derivatives of \( f(x) = (1+x)^{\alpha} \) determine the coefficients for the polynomial terms. This process involves calculating successive derivatives and evaluating each at the point of approximation (in this case, \( x = 0 \)).

• The first derivative of \( f(x) \) is \( f'(x) = \alpha (1+x)^{\alpha-1} \).
• The second derivative is \( f''(x) = \alpha(\alpha - 1)(1+x)^{\alpha-2} \).
• Subsequent derivatives follow a similar pattern, decreasing the power in the expression.

Evaluating these derivatives at \( x = 0 \) gives us the coefficients for the polynomial: \( f(0) = 1 \), \( f'(0) = \alpha \), \( f''(0) = \alpha(\alpha - 1) \), and so on. Each coefficient corresponds to a term in the Taylor series, allowing us to build an accurate polynomial representation of the function near this point.

Radius of convergence

The radius of convergence is a vital factor that determines the interval within which the Taylor series approximates the function accurately. For the series of \( (1+x)^{\alpha} \), as provided in the exercise, it converges for \( x \in (-1,1) \).
This concept is linked to the remainder term, where the series' accuracy improves as the number of terms increases. The larger the radius, the wider the interval over which the series is a good approximation of the function. In practice, knowing the radius of convergence helps us understand where the polynomial is a reliable substitute for the original function.
Convergence is crucial, not just for mathematical elegance, but for ensuring that approximations lead to meaningful and correct predictions. In cases outside the radius of convergence, the Taylor series may diverge and lead to incorrect results.

Remainder term

The remainder term, represented as \( R_{n+1}(x) \), is what makes the Taylor series approximation of a function incomplete. In an ideal scenario, the error between the polynomial approximation and the actual function equals zero. However, practically there is always a small error dictated by the remainder term.
For any finite Taylor polynomial, \( R_{n+1}(x) \) accounts for the missing higher order terms. It determines how closely the Taylor series approximates the function within its convergence radius. As more terms are included, \( R_{n+1}(x) \) approaches zero within the interval of convergence, making the approximation more precise.

• This term ensures that even though we stop calculating after a few derivatives, the series still converges to the actual function over specified intervals.

In summary, the remainder term is critical for assessing the accuracy of the approximation and guarantees that our polynomial mirrors the true function's behavior tightly within its convergence boundary. Understanding and estimating the remainder is crucial, especially in applications requiring precise computations.