91Ó°ÊÓ

A geometric series is a series of the form

• \( a + ar + ar^2 + ar^3 + \cdots \)

where \( a \) is the first term, and \( r \) is the common ratio. The series is particularly useful in mathematics because it can be easily summed up using a simple formula. The sum of an infinite geometric series can be written as

• \( \frac{a}{1 - r} \)

provided that the absolute value of \( r \) is less than one (\( |r| < 1 \)). In the original exercise, the function \( f(x) = \frac{x}{x^2 + 16} \) was rewritten to take advantage of this concept. By manipulating the denominator to resemble the form \( 1 - r \), it aligns with the structure of a geometric series, permitting a series expansion.

Partial sums are a way to approximate the sum of a series by summing only a finite number of terms. If you have a series such as

• \( a_1 + a_2 + a_3 + \cdots \)

its partial sum \( s_n \) consists of the first \( n \) terms:

• \( s_n = a_1 + a_2 + a_3 + \cdots + a_n \)

Partial sums are crucial for understanding how a series behaves, particularly in observing the convergence. In the solution, partial sums of the power series for \( f(x) \) are graphed. These graphs help visualize how the addition of more terms improves the approximation of the function, showing the effectiveness of partial sums in representing the function with fewer terms.

Convergence refers to whether the sequence of partial sums of a series approaches a particular value as more terms are added. If the sequence gets closer to a certain value, the series is said to converge to that value. Different series have different convergence criteria.For geometric series, convergence happens when

• \( |r| < 1 \)

which assures that the series approaches a finite limit. In the context of the exercise, when plotting the function and its partial sums, we observe convergence in real-time as the partial sums become better approximations with the increase of terms \( n \). This practical visualization demonstrates how increasing the number of terms enhances the series representation's accuracy over more points in its domain.

A power series is an infinite series of the form

• \( \sum_{n=0}^{} a_n (x - c)^n \)

where each term is multiplied by a power of \( x - c \). Power series are powerful tools for expressing functions, especially in calculus and analysis. They allow the approximation of complex functions with polynomials.In the exercise solution, the function \( f(x) = \frac{x}{x^2 + 16} \) is expanded into a power series by converting it into a geometric series first. Then multiplying the geometric series by a polynomial leads to the power series representation. The ability to represent functions in this manner aids in simplifying calculations and understanding the behavior of functions over various intervals.