Problem 36 - 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

First three nonzero terms are \(x^3 - 2x^5\). The series converges for all real \(x\).

Step 1: Find the Maclaurin Series for Sine

The Maclaurin series for the sine function is given by: \[sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\]We will use the first few terms of this series to find \(\sin^2(x)\).

Step 2: Calculate the Maclaurin Series for Sine Squared

To find \(\sin^2(x)\), square the Maclaurin series:\[(x - \frac{x^3}{3!} + \cdots)^2 = x^2 - 2x^4 + \cdots\]We square the first term as \(x^2\) and the second term \(-2x^4\), then neglect higher order terms for this calculation.

Step 3: Find the Maclaurin Series for \(f(x) = x \sin^2(x)\)

Using the series for \(\sin^2(x)\), multiply by \(x\):\[ x \cdot \sin^2(x) = x \cdot (x^2 - 2x^4 + \cdots) = x^3 - 2x^5 + \cdots\]The first three nonzero terms for \(f(x)\) are \(x^3 - 2x^5\).

Step 4: Determine the Interval of Convergence

The original Maclaurin series \(\sin(x)\) converges for all real numbers \(x\). Therefore, the series for \(f(x) = x \sin^2(x)\) also converges for all \(x\), implying absolute convergence for all real numbers.

Key Concepts

Sine Function

The sine function, denoted as \(\sin(x)\), is a fundamental trigonometric function. It describes the proportion of the opposite side to the hypotenuse in a right-angled triangle. It is periodic, with a cycle repeating every \(2\pi\) radians. This periodic nature is essential in modeling waves and oscillatory phenomena.
The Maclaurin series is an expansion of \(\sin(x)\) around the point \(x = 0\). This series helps in approximating \(\sin(x)\) using polynomials. Here’s the initial part of \(\sin(x)\)'s Maclaurin series:

• \(x - \frac{x^3}{3!}\)
• \( + \frac{x^5}{5!} - \cdots\)

Each term involves an odd power of \(x\) with alternating signs.

This series showcases how the function can be broken into simpler polynomial terms, making complex calculations more manageable and efficient.

Series Convergence

Convergence in series refers to how a series approaches a specific value as more terms are added. A series converges when the sum of its terms tends to a particular value as you include more and more terms. In mathematical terms, a series is said to converge if its partial sums become closer and closer to a specific limit.
For the Maclaurin series of \(\sin(x)\), the series converges to \(\sin(x)\) for all values of \(x\). Even after truncating the series to a few terms, the resultant polynomial closely approximates \(\sin(x)\) around \(x=0\).

This convergence property is crucial because it allows for precise approximations of functions. By examining the convergence of these series, mathematicians can ensure that their approximations yield reliable results in practical applications.
Especially in our solution, when dealing with recursive operations like squaring the series (in \(\sin^2(x)\)), convergence assured us that our polynomial manipulations still represented the original function within a specific accuracy.

Interval of Convergence

The interval of convergence refers to the set of \(x\) values for which a series converges to a finite number. It’s paramount in ensuring the reliability of series approximations for those \(x\) values.
For the Maclaurin series of \(\sin(x)\), the interval is all real numbers. This wide range ensures that we can use the series for practically any \(x\) to approximate \(\sin(x)\). Similarly, for \(f(x) = x \sin^2(x)\), the interval of convergence spans all real numbers.

This trait of converging across all real numbers makes functions like \(\sin(x)\) invaluable in various fields, from engineering to physics, where precise calculations are pivotal. Such properties make the sine function's series versatile, giving it considerable utility in both theoretical and applied contexts.