Problem 38 - Thomas Calculus

Question

Find the first three nonzero terms of the Maclaurin series for each function and the values of \(x\) for which the series converges absolutely. \(f(x)=x \sin ^{2} x\)

Step-by-Step Solution

Answer

The first three nonzero terms are \(x^3 - \frac{2}{3}x^5\). The series converges absolutely for all real \(x\).

Step 1: Identify the Maclaurin Series Components

The Maclaurin series is a Taylor series expansion of a function about 0. First, recognize that we need to find the series for each individual part of the function before multiplying. We need the Maclaurin series for both the function components: \(x\) and \(\sin^2(x)\).

Step 2: Expand \(\sin(x)\) and \(\sin^2(x)\)

The Maclaurin series for \(\sin(x)\) is: \[ \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \] To find \(\sin^2(x)\), use the identity \(\sin^2(x) = (\sin(x))^2\), then square the first few terms of the \(\sin(x)\) Maclaurin series to approximate \(\sin^2(x)\). After squaring up to \(x^3\), it gives: \[ \sin^2(x) \approx x^2 - \frac{2}{3!}x^4 + \cdots \]

Step 3: Multiply the Series of \(x\) and \(\sin^2(x)\)

Since \(f(x) = x \sin^2(x)\), multiply \(x\) by the expanded series of \(\sin^2(x)\): \[ x(x^2 - \frac{2}{3!}x^4 + \cdots) = x^3 - \frac{2}{3}x^5 + \cdots \] This provides the series as \(x^3\), \(-\frac{2}{3}\), and higher-order terms.

Step 4: Identify the First Three Nonzero Terms

The first three nonzero terms of the expansion are \(x^3\) and \(-\frac{2}{3}x^5\). Further multiplication does not give additional nonzero terms at this stage, hence, the first three nonzero terms are \(x^3\), \(-\frac{2}{3}x^5\).

Step 5: Determine Convergence

The Maclaurin series generally converges for values of \(x\) such that the original function is well-behaved (i.e., \(\sin(x)\) and multipliers provide no natural restriction). For \(\sin(x)\), the series converges for all \(x\). Hence, \(x \sin^2(x)\) converges for all real \(x\).

Key Concepts

Taylor series

The Taylor series is a powerful mathematical tool used to approximate functions as infinite sums of their derivatives at a single point. For a function to be expressed as a Taylor series around a point, it must be infinitely differentiable in the neighborhood of that point.

• The general form of a Taylor series for a function \(f(x)\) around a point \(a\) is given by:

\[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 + \frac{f'''(a)}{6}(x-a)^3 + \cdots\]It allows us to approximate \(f(x)\) using a few terms, which is especially useful for complex functions.For the Maclaurin series, which is a special case of the Taylor series centered at zero, the formula simplifies as the point \(a\) is zero.

series expansion

Series expansion is closely related to the concept of a Taylor series. It refers to expressing a function as an infinite series, often aiding in simplifying calculations and problem-solving.

• The essence of series expansion is to break down functions into polynomial-like entities that are easier to evaluate or integrate.
• While Taylor series provide a method for generating series expansions, other types such as Fourier Series exist for different contexts (e.g., periodic functions).

In our current context, we use series expansion to approximate \( \sin(x) \) and subsequently \( \sin^2(x) \), achieving a form that is multiplicatively useful with \( x \) in the function \( f(x) = x \sin^2(x) \).

convergence of series

Understanding convergence of series is crucial in ensuring that our series approximations are valid representations of functions.

• A series converges if the sequence of its partial sums approaches a specific number, allowing the series to adequately model the function within its interval of convergence.
• For our Maclaurin series, because the sine function is well-behaved and continuous, the series converges for all real \(x\).
• This assurance means our derived series for \(x \sin^2(x)\) accurately reflects the function across its domain.

For students, recognizing this ensures they apply the series only in domains where it reliably converges to the functions they approximate.

trigonometric series

Trigonometric series have their unique role, especially in functions involving periodic elements like sine and cosine.

• For functions like \( \sin(x) \) and transformations thereof (\( \sin^2(x) \)), trigonometric series become particularly relevant.
• These series translate complex wave-like behaviors into manageable polynomials, as seen in our Maclaurin expansion.
• The nature of these series also helps in deducing properties such as periodicity and amplitude in longer expansions.

By expressing \( \sin(x) \) as an infinite series, we lay foundational groundwork for practical use in further scientific calculations and complex function manipulation.

polynomial approximation

Polynomial approximation is a pivotal technique in mathematics, enabling us to simplify functions into manageable forms. This is especially true when direct computation might be too challenging or impossible.

• Using polynomial approximations, we can represent complicated functions over an interval using a degree that suits our precision needs.
• In the case of \( f(x) = x \sin^2(x) \), our result approximates this complex expression using a polynomial derived from expanding \( \sin(x) \).

The first three terms derived, such as \( x^3 \) and \(-\frac{2}{3}x^5\), show how polynomial approximation helps in simplifying expressions by maintaining functional characteristics within a tolerable error range.