Problem 71 - Calculus: Early Transcendentals

Question

Use multiplication of division of power series to find the first three nonzero terms in the Maclaurin series for each function. \( y = (\arctan x)^2 \)

Step-by-Step Solution

Answer

The first three nonzero terms are: \( x^2, - \frac{2}{3}x^4 \).

Step 1: Write the Maclaurin Series for arctan(x)

The Maclaurin series for \( \arctan(x) \) is given by \( \arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \). We want terms up to at least \( x^2 \) in the final series, but we'll need higher powers during intermediate calculations.

Step 2: Square the Series

To find \( (\arctan(x))^2 \), we square the series for \( \arctan(x) \). Instead of directly expanding, consider initial terms: \[(\arctan(x))^2 = (x - \frac{x^3}{3} + \cdots)^2\]Using the distributive property gives us: \[= x^2 - 2 \cdot \frac{x^4}{3} + \text{higher order terms}\]

Step 3: Identify Additional Terms

The highest terms generated from initial squaring (even if non-consecuitve) will also contribute. For example: cross terms like \( x \times \frac{-x^3}{3} \) gives \( \frac{-x^4}{3} \), which coincides with part of above result but a full accounting will reveal that once higher terms involve constant adjustments then it'll result into: \[= x^2 - \frac{2x^4}{3} + \cdots\] Note these will balance out simplifying further series entry.

Step 4: Rearrange and Identify First Three Non-zero Terms

We only need the first three non-zero terms in the new series, so the terms are as follows:\( x^2 \), deploy other simplified powers or gathered terms relate to higher expansions, note next arises for direct variable power specification when approximation aims concluding:\[\begin{align*}(\arctan(x))^2 = & x^2 - \frac{2}{3}x^4 + \text{higher order terms}\end{align*}\]

Key Concepts

Power Series

A power series is a way of representing a function as an infinite sum of terms that involve powers of a variable, typically denoted as x. It's like expanding a function into a polynomial of infinite degree. Each term in the series includes a coefficient multiplied by a power of x. Power series are hugely useful in mathematics because they allow us to approximate functions that might be complicated or impossible to express with simpler algebraic expressions.

• Form: Given as the sum of an infinite series: \( c_0 + c_1x + c_2x^2 + c_3x^3 + \cdots \)
• It converges within a certain radius, meaning it provides meaningful results for x values close enough to zero.
• The power series is centered around a specific point, often zero, which in this case makes it a Maclaurin series.

Whether studying calculus or physics, understanding power series helps in finding approximate values of functions, solving differential equations, and even in developing Taylor series for more complex functions.

arctan(x)

The function \( \arctan(x) \) is the inverse of the tangent function, which is denoted as \( \tan^{-1}(x) \). It's used commonly in trigonometry and calculus to determine the angle whose tangent is a given number. In terms of series, the Maclaurin series for \( \arctan(x) \) gives us a way to write this function as an infinite polynomial.

• The series is written as: \( \arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \)
• This series converges for \( |x| \leq 1 \) and gives a precise calculation whenever x is small.
• The alternating signs in the series are due to the nature of the function and result in a slower convergence for x values farther from zero.

Using this series helps us to understand how values of \( \arctan(x) \) are constructed from basic polynomial terms, making it easier to work with mathematically.

Series Multiplication

Series multiplication involves taking two series and multiplying them together term by term to form a new series. This concept is crucial when looking to square a function such as \( (\arctan(x))^2 \). Essentially, series multiplication follows the same logic as multiplying two polynomials.

• Each term in one series is multiplied by each term in another series.
• Combine like terms, meaning terms with the same power of x are summed together.
• In the context of \( (\arctan(x))^2 \), this results in squaring the Maclaurin series of \( \arctan(x) \).

What makes series multiplication even more fascinating is the discipline it introduces in balancing simplicity with accuracy. We aim to keep the initial terms simple while ensuring all consequential orders are taken into account for correctness, especially when finding only a few leading terms of a power series.

Higher Order Terms

In the context of power series, higher order terms refer to the terms that involve powers of x greater than what is needed for an approximation. When squaring a series, as in the problem \( (\arctan(x))^2 \), higher order terms like \( x^3, x^4, ... \) emerge in the calculations.

• These terms can often be neglected when you're looking for a rough approximation or only a few significant terms.
• However, it's crucial to account for them correctly in initial calculations to ensure the accuracy of lower power terms.
• In expressions such as \( x^2 - \frac{2}{3}x^4 + \dots \), higher terms become relevant for precise applications when accuracy of approximate value is questioned.

Understanding which terms are higher order is essential, especially when balancing between precision and simplicity, to convey the substantial behavior of the function without unnecessary complexity.