Problem 11 - Advanced Engineering Mathematics

Question

In Problems 1-12, expand the given function in a Maclaurin series. Give the radius of convergence of each series. $$ f(z)=\sin z^{2} $$

Step-by-Step Solution

Answer

The Maclaurin series for \( \sin(z^2) \) is \( \sum_{n=0}^{\infty} (-1)^n \frac{z^{4n+2}}{(2n+1)!} \) with an infinite radius of convergence.

Step 1: Recall the Maclaurin Series for Sine

The Maclaurin series for a function is given by \( f(z) = \sum_{n=0}^{fty} \frac{f^{(n)}(0)}{n!} z^n \). For the function \( f(z) = \sin z \), the series is \( \sin z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n+1}}{(2n+1)!} \).

Step 2: Substitute \( z^2 \) for \( z \)

To find the Maclaurin series for \( f(z) = \sin(z^2) \), substitute \( z^2 \) in place of \( z \) in the sine series: \( \sin(z^2) = \sum_{n=0}^{\infty} (-1)^n \frac{(z^2)^{2n+1}}{(2n+1)!} \).

Step 3: Simplify the Series

Simplify the series obtained in the previous step using powers of \( z \): \[ \sin(z^2) = \sum_{n=0}^{\infty} (-1)^n \frac{z^{4n+2}}{(2n+1)!} \]. This series represents the Maclaurin expansion of \( \sin(z^2) \).

Step 4: Determine the Radius of Convergence

The sine function's series has an infinite radius of convergence. Since \( \sin(z^2) \) is simply \( \sin(z) \) with \( z^2 \), the series will also have an infinite radius of convergence. Hence, the series converges for all \( z \).

Key Concepts

Radius of Convergence

The radius of convergence is the range within which a power series reliably converges to a function. When a series converges, it means the sum of its terms approaches a finite value as more terms are added. For the function \( f(z) = \sin(z^2) \), we observe that it is based on the sine function, \( \sin(z) \), which has an infinite radius of convergence. This implies that the series converges for any value of \( z \).

This is a special property of the sine function, owing to the fact that polynomials involved grow slower than exponential terms. And thus, the series smoothly approaches the actual value of \( f(z) = \sin(z^2) \) for any real or complex number. Therefore, no constraints exist on \( z \) for the series to converge.

Sine Function

The sine function is a fundamental trigonometric function with a very regular and repeating pattern. Commonly denoted as \( \sin(z) \), it maps any angle \( z \) to a ratio, commonly appearing in periodic phenomena such as waves.

In the context of a Maclaurin series, the sine function can be expressed as an infinite series:

• \( \sin z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n+1}}{(2n+1)!} \)

This series represents the sine wave as a sum of an infinite number of terms. Each term includes powers of \( z \) and factorials in the denominator, allowing the sine of any angle to be approximated with high precision by truncating after a few terms depending on the required accuracy.

Infinite Series

An infinite series is a sum of an endless number of terms. Fundamental to calculus and analysis, infinite series help in representing functions and curves that cannot be expressed using a finite number of operations.

In infinite series like the Maclaurin series for \( \sin(z^2) \), each additional term makes the series more accurate.
The series:
\[\sin(z^2) = \sum_{n=0}^{\infty} (-1)^n \frac{z^{4n+2}}{(2n+1)!}\]
is an example where powers and factorials control growth, ensuring convergence. Many real-world calculations utilize these series to estimate complex functions across various intervals.

Power Series Expansion

A power series is a way of describing a function as an infinite sum of terms, each involving increasing powers of the variable. Power series expansions are powerful because they can be used to approximate functions to any desired level of accuracy.

In this context, the Maclaurin series is a type of power series centered at zero. To expand a sine function, such as \( \sin(z^2) \), we substitute \( z^2 \) for \( z \) in the standard sine series:

• The resulting expansion is \( \sin(z^2) = \sum_{n=0}^{\infty} (-1)^n \frac{z^{4n+2}}{(2n+1)!} \)

The power series expansion enables complex functions to be represented more simply, allowing for easier computation and analysis.