Problem 23 - Thomas Calculus

Question

Find the Maclaurin series for the functions. \(x \sin x\)

Step-by-Step Solution

Answer

The Maclaurin series for \(x \sin x\) is \(x^2 - \frac{x^4}{3!} + \frac{x^6}{5!} - \cdots\).

Step 1: Understanding the Maclaurin Series

The Maclaurin series for a function is a Taylor series expansion about 0. It is given by the formula: \[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \ldots \] In this problem, we need to find the Maclaurin series for \(f(x) = x \sin x\).

Step 2: Separate Functions: Constants and Operations

Notice that \(x \sin x = x \cdot \sin x\). We can start by finding the Maclaurin series for \(\sin x\) alone, because it is a well-known series, and then multiply by \(x\).

Step 3: Maclaurin Series for \(\sin x\)

The known Maclaurin series for \(\sin x\) is \[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \] This is an alternating series involving odd powers of \(x\).

Step 4: Multiply \(x\) by the Sin Series

Multiply each term of the \(\sin x\) series by \(x\). This results in \[ x \sin x = x(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots) = x^2 - \frac{x^4}{3!} + \frac{x^6}{5!} - \frac{x^8}{7!} + \cdots \] Every term's power increases by 1.

Step 5: Conclusion

Thus, the Maclaurin series for the function \(x \sin x\) is \[ x^2 - \frac{x^4}{3!} + \frac{x^6}{5!} - \frac{x^8}{7!} + \cdots \]

Key Concepts

Taylor series

Imagine a tool that helps us write a function as an infinite sum of terms, making complex functions manageable and simple to study. That’s essentially what the Taylor series does. The Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. Most commonly, this point is 0, and this special case is known as the Maclaurin series.

The formula for a Taylor series about point 'a' is given by: \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots\]This formula essentially tells us that:

• We start with the function value at 'a'.
• Add a term for each derivative of the function, weighted by the power and factorial terms.

When this series is evaluated at x = 0, it becomes the Maclaurin series. It's the same form, but simpler because all terms involving 'a' become zero—resulting in the elegant form many students study.

For a function like \(x \sin x\), understanding how to construct the Maclaurin series gives us a powerful way to approximate the function using polynomial terms. This technique is widely used in calculus and mathematical analysis to approximate functions, making it a fundamental concept for any math student.

sin x series

The sine function is pivotal in trigonometry, and its power series expansion is one of the most familiar. The Maclaurin series for \(\sin x\) is particularly noteworthy for its alternating series of odd powers of \(x\). It is represented as:\[\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\]
This series captures the wave-like nature of the sine function and is calculated using derivatives of \(\sin x\) evaluated at 0.

• The first derivative \(\cos x\) evaluated at 0 gives us 1.
• The second derivative \(-\sin x\) evaluated at 0 gives 0, (hence, no term with \(x^2\)).
• Continuing this pattern, we generate the characteristic odd-powered series.

The series alternates between positive and negative terms. The factorial in the denominator comes from dividing each derivative’s term by its factorial, which also stabilizes and slows the growth of the series terms—essential for convergence.

When tackling \(f(x) = x \sin x\), we utilize this known expansion, managing calculations far simpler by multiplying through by \(x\). This effectively increases the powers of each term in the sine series by 1, resulting in:\[x^2 - \frac{x^4}{3!} + \frac{x^6}{5!} - \cdots\]Understanding the \(\sin x\) series helps in rewriting more complex trigonometric functions efficiently using series expansions.

power series expansion

Power series provides a framework for expressing functions as a sum of powers of \(x\). A power series is simply an infinite sum that looks like this: \[a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots\]Each \(a_n\) is a coefficient that determines how much of each power of \(x\) is included in the series.

Power series are extremely versatile. They not only provide approximations for functions but can also be used to calculate expressions that are otherwise difficult to compute directly. One of the most potent applications of power series is in generating functions, like the Maclaurin and Taylor series, which express more complex mathematical functions as sums of simpler polynomial terms.

For the function \(x \sin x\), writing it as a power series following the steps outlined means you're taking a known power series for \(\sin x\) and adjusting it for an additional factor of \(x\). This results in each term's power being increased by 1:

• Start from the Maclaurin series for \(\sin x\).
• Apply the principle of power series to multiply it by \(x\).
• Watch as each term shifts, creating a new, convergent series that effectively describes \(x \sin x\).

Power series expansions are not just theoretical constructs; they are practical tools used throughout calculus to solve real-world problems and to approximate functions in engineering, physics, and beyond.