Problem 33 - 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. \(f(x)=\cos x-(2 /(1-x))\)

Step-by-Step Solution

Answer

First three nonzero terms: \(-3 - 2x + \frac{x^2}{2}\); converges absolutely for \(|x| < 1\).

Step 1: Maclaurin Series for \(\cos x\)

The Maclaurin series for \(\cos x\) is given by the expansion:\[\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\]The first three nonzero terms are \(1 - \frac{x^2}{2} + \frac{x^4}{24}\).

Step 2: Maclaurin Series for \(\frac{2}{1-x}\)

To find the Maclaurin series for \(\frac{2}{1-x}\), recognize that this is a geometric series:\[\frac{2}{1-x} = 2\left(1 + x + x^2 + x^3 + \cdots\right)\]The first three nonzero terms are \(2 + 2x + 2x^2\).

Step 3: Combine the Series for \(f(x)\)

The function is given by \(f(x) = \cos x - \frac{2}{1-x}\). Thus, we combine the series obtained:\[f(x) = \left(1 - \frac{x^2}{2} + \frac{x^4}{24}\right) - \left(2 + 2x + 2x^2\right)\]Simplify to find the first three nonzero terms:\[-1 - 2x - \left(2 - \frac{x^2}{2}\right) = -1 - 2x - 2 + \frac{x^2}{2} = -3 - 2x + \frac{x^2}{2}\]Thus, the first three nonzero terms are \(-3 - 2x + \frac{x^2}{2}\).

Step 4: Determine Convergence for \(\frac{2}{1-x}\)

The series \(\frac{2}{1-x}\) converges when \(|x| < 1\), since it is a geometric series.

Step 5: Determine Convergence for \(\cos x\)

The series for \(\cos x\) converges for all real \(x\) because it is an entire function.

Step 6: Combine Convergence Conditions

For the function \(f(x)\), we need to satisfy both convergence conditions. The series will only converge absolutely where both components do, which is when \(|x| < 1\).

Key Concepts

Geometric Series

A geometric series is a series where each term is a constant multiple of the previous term. These series are quite versatile and appear in numerous mathematical contexts. Here's how they work:

• The standard form of a geometric series is given by: \[ a + ar + ar^2 + ar^3 + \cdots \] where \(a\) is the first term and \(r\) is the common ratio.
• A geometric series converges when the absolute value of \(r\) is less than 1, written as \(|r| < 1\). This is a crucial condition for determining the behavior of infinite series.
• For the function \( \frac{2}{1-x} \), it can be expanded as a geometric series: \[ 2 \left(1 + x + x^2 + x^3 + \cdots \right) \] with the first term 2 and the common ratio \(x\).
• The series will converge when \(|x| < 1\), capturing a range where the function behaves nicely and the terms continue without becoming infinitely large.

Convergence

Convergence in the context of series is a fundamental concept that helps us understand when a series sums up to a finite value. It's an essential part of working with series like the ones introduced in Maclaurin expansions.

• A series converges if the sum of its terms approaches a specific number as more terms are added.
• For series like those in the Maclaurin expansion, knowing convergence helps us know for which x-values the series will give meaningful results.
• The series \(\cos x\) converges for all real \(x\). This means it provides valid approximations for any real number input.
• The series \(\frac{2}{1-x}\), being a geometric series, converges specifically for \(|x| < 1\). This limitation on \(x\) ensures that the terms do not increase without bound.
• When combining series like in \(f(x) = \cos x - \frac{2}{1-x}\), we must consider the strictest conditions. Hence, the combined series converges where both individual series do, namely when \(|x| < 1\).

Cosine Function

The cosine function, represented as \(\cos x\), is a fundamental trigonometric function with properties conducive to series expansions like the Maclaurin series.

• The Maclaurin series for \(\cos x\) is derived from its Taylor series, centered at 0. It is expressed as: \[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \]
• This expansion is infinite, but we often only need a few terms for practical approximations. Here, the first three nonzero terms are: \(1 - \frac{x^2}{2} + \frac{x^4}{24}\).
• These approximations are highly useful in physics, engineering, and other fields, allowing complex sinusoidal waveforms to be simplified.
• As an entire function, \(\cos x\) converges for all real numbers. Thus, its series form is always applicable, regardless of the input value for \(x\).
• While constructing functions like \(f(x) = \cos x - \frac{2}{1-x}\), understanding the series expansion of \(\cos x\) allows the function's behavior and convergence properties to be analyzed deeply.