Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.
sin x – x + (x³ / 3!)

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Recall the Taylor series expansion of \( \sin x \) at \( x = 0 \), which is given by: \[ \sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]

Write down the given expression explicitly: \[ \sin x - x + \frac{x^3}{3!} \]

Substitute the Taylor series for \( \sin x \) into the expression: \[ \left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \right) - x + \frac{x^3}{3!} \]

Combine like terms by grouping powers of \( x \): Notice that \( x \) and \( -x \) cancel out, and \( -\frac{x^3}{3!} \) and \( +\frac{x^3}{3!} \) also cancel out, leaving the series starting from the \( x^5 \) term:

Write the simplified Taylor series starting from the \( x^5 \) term: \[ \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \cdots \]

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Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point, often x = 0 (Maclaurin series). It approximates functions locally and is essential for expressing functions like sin x as power series.

Power Series Operations

Power series operations include addition, subtraction, multiplication, and differentiation of series. These operations allow manipulation of known series to find new series representations, such as combining or modifying the Taylor series of sin x to match the given expression.

Maclaurin Series for sin x

The Maclaurin series for sin x is an alternating series of odd powers of x with factorial denominators: sin x = x - x³/3! + x⁵/5! - ... . Understanding this series is crucial to simplifying or adjusting terms like sin x – x + (x³/3!) in the problem.

Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.sin x – x + (x³ / 3!)