Problem 18 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The Taylor series for \(x^2 \cos(x^2)\) at \(x=0\) is \(x^2 - \frac{x^6}{2} + \frac{x^{10}}{24} + \ldots\).

Step 1: Identify the Taylor series of cosine

First, recall the Taylor series expansion of the cosine function at zero. The cosine function has the following expansion around zero:\[\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}\]This means that \(\cos(x)\) can be expanded into an infinite series involving powers of \(x\).

Step 2: Substitute and simplify the expression

Next, substitute \(x^2\) into the cosine function from the Taylor series expansion. Given that we need \(\cos(x^2)\), substitute \(x\) with \(x^2\) in the series:\[\cos(x^2) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} (x^2)^{2n}\]This simplifies to:\[\cos(x^2) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{4n}\]This step creates the series for \(\cos(x^2)\).

Step 3: Multiply by \(x^2\)

The initial function given is \(x^2 \cos(x^2)\). Multiply the entire series found in Step 2 by \(x^2\):\[x^2 \cos(x^2) = x^2 \cdot \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{4n}\]This becomes:\[x^2 \cos(x^2) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{4n+2}\]This step combines the series for \(\cos(x^2)\) with the \(x^2\) multiplier.

Step 4: Write out terms of the series

Write out the first few terms of the resulting series from Step 3 to better understand its structure:- For \(n=0\): \(\frac{(-1)^0}{0!} x^2 = x^2\)- For \(n=1\): \(\frac{(-1)^1}{2!} x^6 = -\frac{x^6}{2}\)- For \(n=2\): \(\frac{(-1)^2}{4!} x^{10} = \frac{x^{10}}{24}\)So, the series looks like:\[x^2 - \frac{x^6}{2} + \frac{x^{10}}{24} + \ldots\]

Key Concepts

Power Series

A power series is a fancy way of expressing a function as an infinite sum of terms, where each term is a power of some variable, usually denoted as \(x\). Just imagine a series of numbers, but instead of just numbers, they are attached to powers of \(x\). These series can be super helpful for approximating functions or even calculating values like those of trigonometric functions, exponentials, and logarithms.

Here are some key features of power series:

• They usually look like: \(a_0 + a_1x + a_2x^2 + a_3x^3 + \, \ldots\)
• Each term involves a coefficient \(a_n\) and a power \(x^n\).
• Power series can be used to represent functions by using the idea of convergence (making sure the series doesn’t go off to infinity).

Power series are useful because they give us a way to approximate functions with just a few terms, especially when \(x\) is close to a particular value, known as the center of the series. This makes them a cornerstone in calculus, particularly when calculating Taylor series.

Cosine Function

The cosine function is one of the fundamental functions in trigonometry. It's periodic, meaning it repeats its values in a regular pattern, and is crucial for describing oscillations and waves.

Taylor Series of Cosine

The Taylor series gives us a neat way to express the cosine function as an infinite series. Specifically, at \(x = 0\), it is:\[\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}\]This series makes it easy to calculate \(\cos x\) for small values of \(x\), as only a few terms are needed to get a good approximation. When we substitute \(x^2\) for \(x\) in this series, we can easily find the series for \(\cos(x^2)\), which is immensely helpful when working with transformed trigonometric functions.

Characteristics of the Cosine Series

• The series includes only even powers of \(x\).
• The sign of the terms alternates, providing both positive and negative terms.
• This alternation helps the series quickly converge to the function’s true value.

Infinite Series

An infinite series is just what it sounds like – a sum of an infinite number of terms. This concept is central to calculus and analysis, as it helps in describing functions that might not be expressible using elementary functions alone.

Characteristics of Infinite Series

• Typically written using the summation symbol, \(\sum\).
• The series can be finite or infinite; here, we focus on the latter.
• The main idea is convergence – we analyze whether adding up all the terms leads to a finite number or drifts off to infinity.

An infinite series' behavior is largely dictated by its terms' structure. For example, the Taylor series for \(x^2 \cos(x^2)\) reveals how just a few leading terms of this infinite series can approximate the function well for small \(x\). By carefully choosing and lining up terms starting from \(n=0\) as done in the solution, we see how infinite series are not just abstract constructions, but practical tools to solve real-world problems.