Problem 28 - Thomas Calculus

Question

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=1 /(1-x)^{3}, \quad a=0\)

Step-by-Step Solution

Answer

The Taylor series is \( \sum_{n=0}^{\infty} \frac{n(n+1)}{2} x^n \).

Step 1: Identify the General Form for Taylor Series

The Taylor series of a function \( f(x) \) about \( x = a \) is given by: \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \] For this exercise, we need to find the series about \( x = 0 \) (hence \( a = 0 \)).

Step 2: Find the First Few Derivatives

First calculate a few derivatives of \( f(x) = \frac{1}{(1-x)^3} \):- \( f(x) = (1 - x)^{-3} \)- \( f'(x) = 3(1-x)^{-4} \)- \( f''(x) = 12(1-x)^{-5} \)- \( f'''(x) = 60(1-x)^{-6} \)

Step 3: Evaluate Derivatives at \( x = 0 \)

Substitute \( x = 0 \) into each derivative:- \( f(0) = (1-0)^{-3} = 1 \)- \( f'(0) = 3(1-0)^{-4} = 3 \)- \( f''(0) = 12(1-0)^{-5} = 12 \)- \( f'''(0) = 60(1-0)^{-6} = 60 \)

Step 4: Construct the Taylor Series

Substitute the derivatives evaluated at \( x = 0 \) into the Taylor series formula: \[ f(x) = 1 + 3x + \frac{12}{2!}x^2 + \frac{60}{3!}x^3 + \cdots\]Simplifying each term gives: \[ f(x) = 1 + 3x + 6x^2 + 10x^3 + \cdots\]

Step 5: Recognize the Pattern and Generalize

Notice the coefficients \(1, 3, 6, 10, \cdots\) which are the so-called "triangular numbers." The general pattern is given by the formula for the nth triangular number: \( T_n = \frac{n(n+1)}{2} \).Thus, the Taylor series can be written as:\[ \sum_{n=0}^{\infty} T_n x^n = \sum_{n=0}^{\infty} \frac{n(n+1)}{2} x^n \] where \( T_n \) is the nth triangular number.

Key Concepts

Triangular Numbers

Triangular numbers are a sequence of numbers that form an equilateral triangle when arranged in a dot pattern. To visualize, consider lining up dots to form successive rows, with each row containing one more dot than the previous. For example, the first few triangular numbers are:

• 1 (a single dot)
• 3 (one row of two dots above a single dot)
• 6 (one row of three dots above a row of two dots above a single dot)

The nth triangular number can be calculated using the formula:\[ T_n = \frac{n(n+1)}{2} \]This formula gives the total count of dots that can form a perfect triangle consisting of n rows. Not only a fun visualization in geometry, but triangular numbers also appear in various mathematical contexts, including the Taylor series where they represent coefficients.

Derivatives

Derivatives are fundamentally about finding the rate of change of a function. In calculus, when we speak about a derivative, we refer to the slope of a function at any given point. Mathematically, the derivative of a function \(f(x)\) is denoted as \(f'(x)\).
Understanding derivatives involves:

• The Power Rule: If \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).
• The Chain Rule: For composite functions, the derivative is the product of the outer function's derivative and the derivative of the inner function.

For the function \( f(x) = \frac{1}{(1-x)^3} \), derivatives help in forming the Taylor series by calculating successive rates of change and evaluating them at a specific point, in this case, \(x = 0\). Derivatives unlock the capability to approximate complex functions using polynomials.

Maclaurin Series

A Maclaurin series is a special case of the Taylor series where we expand the function around the point \(x=0\). For a given function \(f(x)\), the Maclaurin series is:\[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots \]Maclaurin series provide a polynomial approximation of the function that can be infinitely extended to represent the function accurately over a wide range of values near zero. This makes them incredibly useful for simplifying complex functions and aiding in computations.When constructing the Maclaurin series for \(f(x) = \frac{1}{(1-x)^3}\), the calculated derivatives are substituted into this series expansion. As a result, each term contributes to building an approximation of the function based on the coefficients derived from the function and its derivatives evaluated at zero.