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A power series is essentially an infinite sum involving powers of a variable, usually denoted as \( x \). The general form of a power series is given by the expression:\[\sum_{n=0}^{fty} a_n x^n\]Here, \( a_n \) are coefficients specific to the particular series in question. Power series can represent functions in a region and help in approximating the behavior of functions near a particular point, commonly around \( x = 0 \), which is also known as the center.

In our exercise, recognizing a function as a geometric series allows us to express it as a power series around \( x = 0 \). For example, the function: - For \( f(x) = \frac{5}{3-x} \): This function can be rewritten to fit the geometric series form, then expressed as \( \sum_{n=0}^{\infty} \frac{5x^n}{3^{n+1}} \).

Power series expansions provide a way to explore functions analytically and can be extremely helpful in calculus, especially when dealing with integrals and derivatives of complex functions.

The interval of convergence is crucial when working with power series, as it indicates the range of \( x \)-values for which the series converges to a finite value. Without defining this interval, the usefulness of a power series is limited.

To determine the interval of convergence, one usually applies the ratio test or another suitable convergence test. For a geometric series of the form \( \frac{a}{1-r} \), the series converges when \( |r| < 1 \).

• For the function \( f(x) = \frac{5}{3-x} \), the geometric ratio is \( \frac{x}{3} \). Setting \( \left| \frac{x}{3} \right| < 1 \) gives the solution \( |x| < 3 \), thus the interval of convergence is \(-3 < x < 3\).
• Similarly, for \( g(x) = \frac{3}{x-2} \), we find the geometric ratio \( \frac{x}{2} \), leading to the interval \(-2 < x < 2\).

These intervals signify where the original function can be accurately expressed as a power series, which is immensely useful for solving calculus problems.

Series expansion refers to expressing a function as an infinite sum, typically involving terms of a variable raised to increasing powers. This technique is pivotal in mathematics as it provides a way to analyze and manipulate functions that may otherwise be cumbersome to work with.

In terms of power series, series expansion turns a function into an infinite polynomial. Once a function is expressed in this manner, it can reveal properties like continuity and differentiability, as well as provide estimates for function values.

The step-by-step solution illustrates series expansion by converting functions like \( \frac{5}{3-x} \) and \( \frac{3}{x-2} \) into sums of an infinite number of terms:- \( f(x) = \frac{5}{3-x} \) becomes \( \sum_{n=0}^{\infty} \frac{5x^n}{3^{n+1}} \).- \( g(x) = \frac{3}{x-2} \) transforms into \(-\sum_{n=0}^{\infty} \frac{3x^n}{2^{n+1}} \).This method is fundamental in many areas, from theoretical physics to engineering.