Problem 20 - Thomas Calculus

Question

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(13-30\) . $$\sin ^{2} x$$

Step-by-Step Solution

Answer

\(\sin^2x = x^2 - \frac{x^4}{3} + \cdots\)

Step 1: Express the Function Using Known Series

The function we need the Taylor series for is \(\sin^2x\). We start by expressing \(\sin x\) as a Taylor series centered at \(x=0\). The Taylor series for \(\sin x\) is given by \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots\).

Step 2: Square the Series

Since we need \(\sin^2x\), we square the Taylor series of \(\sin x\). This means squaring each term of the series and applying the distributive property:\[\sin^2x = \left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\right)^2.\]

Step 3: Compute the Squared Series Terms

Compute the series terms from squaring the series. The first few terms arise from squaring \(x\), \(x^3\), and so on:\[\sin^2x = x^2 - (2 \cdot x \cdot \frac{x^3}{3!}) + \left(\frac{x^3}{3!}\right)^2 + \cdots\] This gives:\[\sin^2x = x^2 - \frac{2x^4}{3!} + \cdots\].

Step 4: Identify Significant Terms

Focus on deriving a few initial terms of the series. Given the complexity of the power expansions, it often suffices to identify several leading terms. Summarizing, we find: \[\sin^2x = x^2 - \frac{x^4}{3} + \cdots\]

Step 5: Verify Alternating Pattern

Having obtained the initial terms, ensure the alternationm matches expected trigonometric expansions for verification. This series shows the expected alternating pattern of even powers.

Key Concepts

Power Series

The Power Series is a crucial mathematical tool that allows us to express functions as an infinite sum of terms. Each term is derived from raising a variable to successive powers and multiplying by constant coefficients. For example, a generic power series looks like this:\[ f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots \]This format is incredibly useful because it simplifies complex functions into more manageable components.
Power series are used to approximate functions, calculate outcomes of complicated expressions, and analyze behavior of functions around specific points. They are the backbone for many mathematical operations, including engineering and physics applications.

• Each term in the series contributes to the overall approximation of the function.
• The more terms you include, the closer the approximation gets to the actual function value.
• In some cases, the power series converges exactly to the function itself within a certain interval.

Understanding power series is key to mastering more advanced math topics like the Taylor and Maclaurin series.

Sin Function

The sine function, denoted as \(\sin x\), is one of the most fundamental trigonometric functions. It is defined for all real numbers and is periodic with a period of \(2\pi\). This means its graph repeats its shape every \(2\pi\) units along the x-axis.
The Taylor series allows us to express \(\sin x\) as an infinite sum of terms:\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]

• This expansion gives us an alternate way to represent \(\sin x\).
• It shows that \(\sin x\) can be broken down into simpler components.
• The series alternates between positive and negative terms, which balances the expression.
• Only odd-powered terms appear, consistent with the properties of the sine function.

The Taylor series for \(\sin x\) converges for all x, meaning this expansion matches the sine function exactly for any value of x. This powerful expansion is used in many fields, from engineering to computer graphics, because it simplifies calculations that involve sine.

Maclaurin Series

The Maclaurin Series is a specific form of Taylor series. It is a representation of a function as an infinite sum of terms calculated from the derivatives of the function at one point, usually \(x = 0\). This makes it a special case of Taylor series with the center at zero.
The generic form of a Maclaurin series is:\[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \]

• The Maclaurin series simplifies calculations by centering them around zero.
• It's just as flexible and powerful as any Taylor series centered at another point.
• Most common functions like \(\sin x\), \(\cos x\), or \(e^x\) have well-known Maclaurin series.
• Having a series expansion can make finding function values or solving differential equations much simpler.

With its grounded base at zero, the Maclaurin series often serves as a starting point for various mathematical applications.

Series Expansion

Series expansion is a method of expressing a more complex function in terms of a series of simpler functions. This is advantageous because it allows better understanding and easier computation in many cases. For example, the expansion of \(\sin^2x\) using power series transforms it into simpler components, which makes calculations straightforward.
The process involves substituting a known series into the function you want to expand and then simplifying it:

• First, you express a complicated function using simpler series elements.
• Then, you apply algebraic operations like squaring or substituting to expand the function.
• The final result gives a new series that represents the original function.
• Series expansions can approximate the function value at certain points or even across defined intervals.

This approach is not only limited to trigonometric functions but can be applied to any differentiable function, making it extremely versatile in both theoretical and practical applications. Knowing how to effectively perform series expansions can provide significant advantages in the manipulation and computation of mathematical functions.